The
Local
Pro-p
Anabelian
Geometry
of
Curves
by
Shinichi
Mochizuki
INTRODUCTION
The
Anabelian
Geometry
of
Grothendieck:
Let
X
be
a
connected
scheme.
Then
one
can
associate
(after
Grothendieck)
to
X
its
algebraic
fundamental
group
π
1
(X).
This
group
π
1
(X)
is
a
profinite
group
which
is
uniquely
determined
(up
to
inner
automorphisms)
by
the
property
that
the
category
of
finite,
discrete
sets
equipped
with
a
continuous
π
1
(X)-action
is
equivalent
to
the
category
of
finite
étale
coverings
of
X.
Moreover,
the
assignment
X
→
π
1
(X)
is
a
functor
from
the
category
of
connected
schemes
(and
morphisms
of
schemes)
to
the
category
of
profinite
topological
groups
and
continuous
outer
homomorphisms
(i.e.,
continuous
homomorphisms
of
topological
groups,
where
we
identify
any
two
homomorphisms
that
can
be
obtained
from
one
another
by
composition
with
an
inner
automorphism).
Now
let
K
be
a
field.
Let
Γ
K
be
the
absolute
Galois
group
of
K.
Then
π
1
(Spec(K))
may
be
identified
with
Γ
K
.
Let
X
K
be
a
variety
(i.e.,
a
geometrically
integral
separated
scheme
of
finite
type)
over
K.
Then
the
structure
morphism
X
K
→
Spec(K)
defines
a
natural
augmentation
π
1
(X
K
)
→
Γ
K
.
The
kernel
of
this
morphism
π
1
(X
K
)
→
Γ
K
is
a
closed
normal
subgroup
of
π
1
(X
K
)
–
called
the
geometric
fundamental
group
of
X
K
–
def
which
may
be
identified
with
π
1
(X
K
)
(where
X
K
=
X
K
⊗
K
K).
If,
moreover,
one
fixes
a
prime
number
p,
then
one
can
form
the
maximal
pro-p
quotient
π
1
(X
K
)
(p)
of
π
1
(X
K
).
Since
the
quotient
π
1
(X
K
)
→
π
1
(X
K
)
(p)
is
characteristic,
it
follows
that
the
kernel
of
this
quotient
is,
in
fact,
a
normal
subgroup
of
π
1
(X
K
).
The
quotient
of
π
1
(X
K
)
by
this
normal
subgroup
will
be
denoted
Π
X
K
.
Thus,
Π
X
K
inherits
a
natural
augmentation
Π
X
K
→
Γ
K
from
that
of
π
1
(X
K
).
Now
let
us
consider
the
assignment
π
1
(−)
K
:
{X
K
→
Spec(K)}
→
{π
1
(X
K
)
→
Γ
K
}
This
assignment
defines
a
functor
from
the
category
C
K
of
K-varieties
(whose
morphisms
are
K-linear
morphisms
of
varieties)
to
the
category
G
K
whose
objects
are
profinite
topo-
logical
groups
equipped
with
an
augmentation
to
Γ
K
,
and
whose
morphisms
are
continu-
ous
outer
homomorphisms
of
topological
groups
that
lie
over
Γ
K
.
It
was
the
intuition
of
Grothendieck
(see
[Groth])
that:
1
For
certain
types
of
K,
if
one
replaces
C
K
and
G
K
by
“certain
appro-
and
G
K
(such
that
π
1
(−)
K
still
maps
C
K
into
priate”
subcategories
C
K
G
K
),
then
π
1
(−)
K
should
be
fully
faithful.
Here,
the
“certain
appropriate”
subcategories
C
K
for
which
this
piece
of
intuition
was
to
hold
true
were
tentatively
assigned
the
appellation
anabelian,
while
the
piece
of
intuition
itself
came
to
be
referred
to
as
Grothendieck’s
Conjecture
of
Anabelian
Geometry
(or,
sim-
ply,
the
“Grothendieck
Conjecture,”
for
short).
Roughly
speaking,
the
sorts
of
varieties
that
were
thought
to
be
likely
to
be
“anabelian”
were
varieties
that
are
“sufficiently
hyper-
bolic.”
(Note
that,
as
one
can
see
in
the
case
of
curves,
hyperbolic
varieties
tend
to
have
highly
nonabelian
fundamental
groups,
hence
the
term
“anabelian.”
In
higher
dimensions,
however,
things
are
not
so
simple
–
see,
e.g.,
[IN].)
A
variant
of
the
above
“profinite”
Grothendieck
Conjecture
is
the
following
“pro-
p”
Grothendieck
Conjecture:
Namely,
instead
of
considering
π
1
(−)
K
,
one
considers
the
functor
that
assigns
to
the
K-variety
X
K
the
augmented
group
Π
X
K
→
Γ
K
.
The
“pro-p”
Grothendieck
Conjecture
then
asserts
that,
in
certain
situations,
this
functor
should
be
fully
faithful.
The
present
paper
is
concerned
with
proving
various
versions
of
the
profinite
and
pro-p
Grothendieck
Conjectures
under
various
conditions.
Statement
of
the
Main
Results:
Let
p
be
a
prime
number.
Let
K
be
a
sub-p-adic
field
(cf.
Definition
15.4
(i)),
i.e.,
a
subfield
of
a
finitely
generated
field
extension
of
Q
p
.
In
this
paper,
we
prove
a
pro-p
version
of
the
Grothendieck
Conjecture
(Theorem
A)
for
dominant
morphisms
between
“smooth
pro-varieties”
and
“hyperbolic
pro-curves”
over
K.
(Here,
by
a
“smooth
pro-variety”
(respectively,
“hyperbolic
pro-curve”)
over
K,
we
mean
a
K-scheme
which
can
be
written
as
a
projective
limit
of
smooth
varieties
(respectively,
hyperbolic
curves)
over
K
in
which
the
transition
morphisms
are
birational
–
cf.
Definitions
15.4
(ii)
and
16.4.)
We
then
give
various
versions
of
Theorem
A
(namely,
Theorems
A
and
A
)
for
truncated
fundamental
groups.
From
Theorem
A,
we
also
derive
a
profinite
version
of
the
Grothendieck
Conjecture
(Theorem
B)
for
morphisms
between
function
fields
(of
arbitrary
dimension)
over
K.
Next,
we
apply
Theorem
A
to
prove
the
“injectivity
part”
(Theorem
C)
of
the
pro-p
“Section
Conjecture.”
Finally,
in
an
Appendix,
we
derive
from
Theorem
A
an
isomorphism
version
of
the
Grothendieck
Conjecture
for
certain
hyperbolic
surfaces
(Theorem
D).
def
Notation:
If
K
is
a
field
and
X
K
is
a
K-scheme,
we
denote
by
Π
prf
X
K
=
π
1
(X
K
)
the
fundamental
group
of
X
K
(for
some
choice
of
base-point),
and
by
Γ
K
the
absolute
Galois
group
of
K.
Then
we
have
a
natural
morphism
Π
prf
X
K
→
Γ
K
whose
kernel
is
the
geometric
prf
prf
fundamental
group
Δ
X
⊆
Π
X
K
.
Let
Δ
X
be
the
maximal
pro-p
quotient
of
Δ
prf
X
.
Then
prf
prf
the
kernel
of
Δ
X
→
Δ
X
is
a
normal
subgroup
of
Π
X
K
,
so
by
forming
the
quotient
of
Π
prf
X
K
by
this
normal
subgroup,
we
obtain
a
group
Π
X
K
,
together
with
a
morphism
Π
X
K
→
Γ
K
def
whose
kernel
is
Δ
X
.
Next,
if
Δ
is
a
topological
group,
we
let:
Δ{0}
=
Δ;
for
i
≥
1,
def
Δ{i}
=
[Δ{i
−
1},
Δ{i
−
1}]
(here
we
mean
“the
closed
subgroup
generated
by
the
purely
2
group-theoretic
commutator
subgroup”).
In
the
specific
case
of
fundamental
groups,
let
us
write
“Π
i
”
(respectively,
“Δ
i
”)
for
Π/Δ{i}
(respectively,
Δ/Δ{i}).
Our
first
main
theorem
is
the
following:
Theorem
A.
Let
p
be
a
prime
number.
Let
K
be
sub-p-adic
(cf.
Definition
15.4
(i)).
Let
X
K
be
a
smooth
pro-variety
over
K.
Let
Y
K
be
a
hyperbolic
pro-curve
over
K.
Let
Hom
dom
K
(X
K
,
Y
K
)
be
the
set
of
dominant
K-morphisms
from
X
K
to
Y
K
.
Let
open
prf
prf
Hom
Γ
K
(Π
X
K
,
Π
Y
K
)
(respectively,
Hom
open
Γ
K
(Π
X
K
,
Π
Y
K
)),
be
the
set
of
open,
continuous
prf
group
homomorphisms
Π
X
K
→
Π
Y
K
(respectively,
Π
prf
X
K
→
Π
Y
K
)
over
Γ
K
,
considered
up
to
composition
with
an
inner
automorphism
arising
from
Δ
Y
(respectively,
Δ
prf
Y
).
Then
the
natural
maps
open
prf
prf
Hom
dom
K
(X
K
,
Y
K
)
→
Hom
Γ
K
(Π
X
K
,
Π
Y
K
)
→
Hom
open
Γ
K
(Π
X
K
,
Π
Y
K
)
are
bijective.
This
Theorem
is
given
as
Theorem
16.5
(cf.
also
the
Remark
following
Theorem
16.5)
in
the
text.
It
is
from
Theorem
A
that
all
of
the
other
major
results
of
this
paper
are
derived.
We
remark
that:
(1)
In
fact,
really,
the
main
portion
of
Theorem
A
is
the
bijectivity
of
the
first
and
third
Hom’s.
That
is
to
say,
the
bijectivity
of
the
first
and
second
Hom’s
follows
formally
from
the
bijectivity
of
the
first
and
third
Hom’s.
See
the
Remark
following
Theorem
16.5
for
more
details.
(2)
The
notion
of
a
“smooth
pro-variety”
(respectively,
“hyperbolic
pro-
curve”)
has
as
special
cases:
(i)
a
smooth
variety
(respectively,
hyper-
bolic
curve)
over
K;
(ii)
the
spectrum
of
a
function
field
of
arbitrary
dimension
(respectively,
function
field
of
dimension
one)
over
K.
(3)
There
exists
a
substantial
body
of
people
who,
when
they
speak
of
“the
Grothendieck
Conjecture”
(respectively,
“the
pro-p
Grothendieck
Conjecture”),
refer
to
the
following
rather
specific
statement
(which
is
a
special
case
of
the
general
philosophy
discussed
in
the
preceding
subsection):
If
K
is
a
finitely
generated
extension
of
Q,
and
X
K
and
Y
K
are
either
hyperbolic
curves
over
K
or
the
spectra
of
one-dimensional
function
fields
over
K,
then
the
isomorphisms
of
X
K
with
Y
K
are
in
natural
bijective
correspondence
with
the
outer
isomorphisms
over
Γ
K
of
prf
Π
prf
X
K
(respectively,
Π
X
K
)
with
Π
Y
K
(respectively,
Π
Y
K
).
This
statement
is
manifestly
a
special
case
of
the
profinite
version
–
see
(1)
above
–
(respectively,
pro-p
version,
i.e.,
version
as
stated
above)
of
Theorem
A.
3
We
also
have
truncated
versions
of
Theorem
A
(Theorems
18.1
and
18.2
in
the
text):
Theorem
A
.
Let
K
be
sub-p-adic.
Let
X
K
be
a
smooth
variety
over
K.
Let
Y
K
be
a
hyperbolic
curve
over
K.
Let
n
≥
5.
Then
every
continuous
open
homomorphism
θ
:
Π
nX
K
→
Π
nY
K
over
Γ
K
induces
a
dominant
morphism
μ
:
X
K
→
Y
K
whose
induced
morphism
on
fun-
damental
groups
coincides
(up
to
composition
with
an
inner
automorphism
arising
from
n−3
Δ
Y
K
)
with
the
morphism
Δ
n−3
X
K
→
Δ
Y
K
defined
by
considering
θ
“modulo
Δ{n
−
3}.”
Let
K
be
sub-p-adic.
Let
X
K
be
a
smooth
pro-variety
over
K.
Let
Y
K
Theorem
A
.
be
a
hyperbolic
pro-curve
over
K.
Let
n
0
be
the
minimum
transcendence
degree
over
Q
p
of
all
finitely
generated
field
extensions
of
Q
p
that
contain
K.
Let
n
0
be
the
transcendence
def
degree
over
K
of
the
function
field
of
X
K
.
Let
n
0
=
n
0
+
2(n
0
−
1)
+
1.
Let
n
≥
3n
0
+
5.
Then
every
continuous
open
homomorphism
θ
:
Π
nX
K
→
Π
nY
K
over
Γ
K
induces
a
dominant
morphism
μ
:
X
K
→
Y
K
whose
induced
morphism
on
fundamental
groups
coincides
(up
to
composition
with
an
inner
automorphism
arising
0
0
→
Δ
n−3−3n
defined
by
considering
θ
“mod-
from
Δ
Y
K
)
with
the
morphism
Δ
n−3−3n
X
K
Y
K
ulo
Δ{n
−
3
−
3n
0
}.”
Our
second
main
theorem
(Theorem
17.1
in
the
text)
is
the
following:
Theorem
B.
Let
p
be
a
prime
number.
Let
K
be
sub-p-adic.
Let
L
and
M
be
function
fields
of
arbitrary
dimension
over
K.
(In
particular,
we
assume
that
K
is
alge-
braically
closed
in
L
and
M.)
Let
Hom
K
(Spec(L),
Spec(M))
be
the
set
of
K-morphisms
from
M
to
L.
Let
Hom
open
Γ
K
(Γ
L
,
Γ
M
)
be
the
set
of
open,
continuous
group
homomorphisms
Γ
L
→
Γ
M
over
Γ
K
,
considered
up
to
composition
with
an
inner
automorphism
arising
from
Ker(Γ
M
→
Γ
K
).
Then
the
natural
map
Hom
K
(Spec(L),
Spec(M))
→
Hom
open
Γ
K
(Γ
L
,
Γ
M
)
is
bijective.
Note
that
in
characteristic
zero,
this
generalizes
the
results
of
[Pop1],
[Pop2],
where
a
similar
result
to
Theorem
B
is
obtained,
except
that
the
morphisms
Γ
L
→
Γ
M
are
required
to
be
isomorphisms,
and
K
is
required
to
be
finitely
generated
over
Q.
4
Our
third
main
theorem
(Theorem
19.1
in
the
text)
is
the
following:
Theorem
C.
Let
p
be
a
prime
number.
Let
K
be
sub-p-adic.
Let
X
K
be
a
hyperbolic
curve
over
K.
Let
X
K
(K)
be
the
set
of
K-valued
points
of
X
K
.
Let
Sect(Γ
K
,
Π
X
K
)
be
the
set
of
sections
Γ
K
→
Π
X
K
of
Π
X
K
→
Γ
K
,
considered
up
to
composition
with
an
inner
automorphism
arising
from
Δ
X
.
Then
the
natural
map
X
K
(K)
→
Sect(Γ
K
,
Π
X
K
)
is
injective.
Finally,
in
the
Appendix,
we
use
Theorem
A
to
derive
the
following
result:
Theorem
D.
Let
p
be
a
prime
number.
Let
K
be
sub-p-adic.
Let
X
K
and
Y
K
be
hyperbolically
fibred
surfaces
(see
Definition
a2.1
in
the
Appendix
for
a
precise
definition
of
this
term)
over
K.
Let
Isom
K
(X
K
,
Y
K
)
be
the
set
of
K-isomorphisms
(in
the
category
prf
of
K-schemes)
between
X
K
and
Y
K
.
Let
Isom
Γ
K
(Π
prf
X
K
,
Π
Y
K
)
be
the
set
of
continuous
prf
group
isomorphisms
Π
prf
X
K
→
Π
Y
K
over
Γ
K
,
considered
up
to
composition
with
an
inner
automorphism
arising
from
Δ
prf
Y
.
Then
the
natural
map
prf
Isom
K
(X
K
,
Y
K
)
→
Isom
Γ
K
(Π
prf
X
K
,
Π
Y
K
)
is
bijective.
Recent
Work
on
the
Grothendieck
Conjecture:
In
this
subsection,
we
would
like
to
take
a
brief
look
at
recent
work
on
the
Grothendieck
Conjecture.
We
will
concentrate
only
on
major
and
recent
(since
the
early
1980’s)
develop-
ments
that
relate
to
the
present
paper,
and
we
make
no
pretense
of
giving
a
complete
history
of
work
on
the
Grothendieck
Conjecture.
Before
beginning,
it
is
worth
pointing
out
that
until
the
appearance
of
the
present
paper
(and
its
earlier
version
[Mzk2]),
it
was
widely
assumed
that
the
base
field
“K”
that
appears
in
the
Grothendieck
Conjecture
should
be
assumed
to
be
finitely
generated
over
Q.
It
is
not
clear
precisely
why
this
came
to
be
as-
sumed
by
most
people
working
in
the
field,
but
one
possible
cause
is
that
the
Grothendieck
Conjecture
appears
to
have
originated
as
an
alternative
approach
to
diophantine
geometry
([Groth],
[Pop3]).
Thus,
if
one’s
ultimate
aim
is
applications
to
diophantine
geometry,
it
is
quite
natural
only
to
look
at
global
fields
(i.e.,
finitely
generated
extensions
of
Q),
and
not
at
p-adic
local
fields,
for
instance.
Another
reason
for
the
fixation
on
finitely
generated
extensions
of
Q
appears
to
have
been
that
many
people
conceived
of
the
Grothendieck
Conjecture
as
an
anabelian
version
5
of
the
Tate
Conjecture
(Faltings’
theorem
–
see
[Falt3])
for
abelian
varieties
over
number
fields.
In
fact,
this
point
of
view
became
so
deeply
engrained
that
it
gave
rise
to
a
tendency
for
people
to
try
to
prove
the
Grothendieck
Conjecture
for
hyperbolic
curves
over
number
fields
by
deducing
it
from
the
Tate
Conjecture
for
abelian
varieties
over
number
fields
(see,
e.g.,
[Naka3]).
In
fact,
even
important
early
work
–
such
as
[Naka1]
on
the
Grothendieck
Conjecture
for
genus
zero
hyperbolic
curves
over
number
fields,
or
[Pop1],
[Pop2]
which
treat
the
birational
case
–
which
does
not
try
to
deduce
the
Grothendieck
Conjecture
from
the
Tate
Conjecture
still
had
a
distinctly
global
flavor,
and
relied
extensively
on
essentially
global
techniques.
These
global
techniques
ultimately
proved
to
be
rather
irrelevant
to
the
proof
of
the
main
results
of
the
present
paper.
Nevertheless,
what
was
important
(from
the
point
of
view
of
the
author)
about
the
work
of
H.
Nakamura
was
that
it
established
a
host
of
basic
techniques
for
studying
the
outer
Galois
action
on
the
(nonabelian)
geometric
fundamental
group
of
a
hyperbolic
curve,
which
was
unknown
territory
to
most
arithmetic
geometers,
who
were
only
familiar
with
the
Galois
action
on
the
abelian
fundamental
group
of
an
abelian
variety.
It
was
this
culture
of
basic
techniques
(due
to
H.
Nakamura)
which
permitted
the
subsequent
development
of
more
powerful
approaches
to
the
Grothendieck
Conjecture
itself,
as
discussed
below.
The
next
important
development
(in
early
1995)
was
the
work
of
A.
Tamagawa
([Tama])
in
which
it
was
shown
that
the
isomorphism
class
of
any
affine
hyperbolic
curve
over
an
absolutely
finitely
generated
field
is
functorially
determined
by
the
outer
Galois
ac-
tion
on
its
profinite
geometric
fundamental
group.
(In
fact,
by
a
relatively
straightforward
argument
([Mzk1]),
it
is
possible
to
remove
the
“affineness”
hypothesis
in
Tamagawa’s
result,
at
least
in
characteristic
zero.)
Unlike
the
results
of
the
present
paper,
Tamagawa’s
result
holds
in
positive
characteristic
as
well
as
in
characteristic
zero.
Moreover,
Tama-
gawa’s
ideas
on
characterizing
those
sections
of
π
1
(X
K
)
→
Γ
K
(where
X
K
is
a
hyperbolic
curve
over
a
field
K)
that
arise
from
geometric
points
of
X
K
played
a
pivotal
role
in
inspir-
ing
the
author
to
prove
the
results
of
the
present
paper.
Concretely
speaking,
the
influence
of
these
ideas
of
Tamagawa
can
be
seen
in
the
argument
of
Section
9
of
the
present
paper.
Finally,
we
turn
to
discussing
the
results
of
the
present
paper
in
a
historical
context.
As
observed
previously,
the
results
of
the
present
paper
are
(unlike
Tamagawa’s
results)
only
valid
in
characteristic
zero.
On
the
other
hand,
the
main
advances
of
Theorem
A
of
the
present
paper
relative
to
[Tama]
are
as
follows:
(1)
The
hyperbolic
curves
involved
are
allowed
to
be
proper.
In
fact,
we
even
allow
what
we
call
hyperbolic
pro-curves.
(2)
Instead
of
dealing
with
profinite
geometric
fundamental
groups,
we
deal
with
pro-p
geometric
fundamental
groups.
Pro-p
results
tend
to
be
stronger
than
profinite
results
in
the
sense
that
profinite
results
usu-
ally
follow
immediately
from
pro-p
results
(see
the
Remark
following
Theorem
16.5
for
more
details).
(3)
Instead
of
just
considering
isomorphisms,
we
allow
arbitrary
dominant
morphisms
between
the
varieties
involved.
6
(4)
The
variety
“on
the
left”
(in
the
“Hom(−,
−)”
of
Theorem
A)
is
allowed
to
be
higher-dimensional,
and
need
not
even
be
hyperbolic.
(5)
The
base
field
can
be
any
subfield
of
a
finitely
generated
extension
of
Q
p
,
whereas
for
Tamagawa
(in
characteristic
zero)
the
base
field
must
be
a
finitely
generated
extension
of
Q.
From
the
point
of
view
of
the
author,
(5)
is
the
most
fundamental
and
important
advance
as
it
is
associated
with
the
fact
that
the
techniques
of
the
present
paper
are
fundamentally
different
from
those
of
both
Nakamura
and
Tamagawa
in
that
they
are
couched
in
the
world
of
p-adic
Hodge
theory.
That
is
to
say,
the
proof
of
Theorem
A
may
be
regarded
as
an
application
of
the
theory
of
[Falt1]
and
[BK].
It
is
precisely
the
use
of
these
p-adic
techniques
that
allowed
the
author
to
prove
a
result
which
was
much
stronger
(in
the
above
five
senses)
than
the
result
of
[Tama]
(in
characteristic
zero)
or
[Mzk1].
Moreover,
it
is
the
opinion
of
the
author
that:
The
reason
that
it
took
so
long
for
Theorem
A
to
be
discovered
was
the
overwhelming
prejudice
of
most
people
in
the
field
that
the
Grothendieck
Conjecture
for
hyperbolic
curves
is
an
essentially
global
result,
akin
to
the
Tate
Conjecture
for
abelian
varieties
over
number
fields.
In
fact,
however,
it
is
much
more
natural
to
regard
the
Grothendieck
Conjecture
for
hyperbolic
curves
as
an
essentially
local,
p-adic
result
that
belongs
to
that
branch
of
arithmetic
geometry
known
as
p-adic
Hodge
theory.
Moreover,
it
is
the
feeling
of
the
author
that,
more
than
the
technical
details
of
the
state-
ment
of
Theorem
A,
it
is
this
fact
–
i.e.,
that
the
Grothendieck
Conjecture
for
hyperbolic
curves
is
best
understood
not
as
a
global,
number-theoretic
result,
but
rather
as
a
result
in
p-adic
Hodge
theory
–
that
is
the
central
discovery
of
this
paper.
The
Structure
of
the
Proof:
First,
we
remark
that
most
of
the
paper
(Sections
1
through
14)
is
devoted
to
proving
the
following
technical
result:
(∗)
tech
Suppose
that
K
is
a
finite
extension
of
Q
p
.
Let
X
K
and
Y
K
be
proper
hyperbolic
curves
over
K.
Let
U
K
be
the
spectrum
of
the
function
field
of
X
K
.
Then
every
continuous
surjective
homomorphism
θ
:
Π
U
K
→
Π
Y
K
over
Γ
K
arises
from
some
geometric
morphism
U
K
→
Y
K
.
Section
15
is
devoted
to
the
rather
standard
technicalities
necessary
to
generalize
(∗)
tech
to
the
case
where
K
is
any
subfield
of
a
finitely
generated
extension
of
Q
p
.
Section
16
is
devoted
to
carrying
out
a
standard
“cutting
by
hyperplane
sections
argument”
which
allows
one
to
replace
U
K
by
a
higher-dimensional
smooth
pro-variety
(which
thus
completes
7
the
proof
of
Theorem
A).
Section
17
is
devoted
to
an
induction
argument
used
to
derive
Theorem
B
from
Theorem
A.
Section
18
discusses
how
the
techniques
used
to
prove
Theorem
A
in
fact
also
give
rise
to
the
truncated
versions
Theorem
A
and
Theorem
A
.
Finally,
Section
19
reviews
a
standard
argument
(already
present
in
the
work
of
H.
Nakamura)
which
allows
one
to
derive
Theorem
C
from
Theorem
A.
Thus,
the
rest
of
this
subsection
will
be
devoted
to
outlining
the
proof
of
(∗)
tech
in
Sections
1
through
14.
In
order
to
simplify
the
discussion,
let
us
consider
first
the
following
slightly
modified
version
of
(∗)
tech
:
(∗)
prop
Suppose
that
K
is
a
finite
extension
of
Q
p
.
Let
X
K
and
Y
K
be
proper
hyperbolic
curves
over
K.
Then
every
continuous
surjective
homomorphism
θ
:
Π
X
K
→
Π
Y
K
over
Γ
K
arises
from
some
geometric
morphism
X
K
→
Y
K
.
(Here,
“prop”
stands
for
“proper”
–
i.e.,
since,
unlike
in
(∗)
tech
,
where
U
K
appeared,
in
(∗)
prop
only
proper
curves
appear.)
Thus,
(∗)
tech
implies
(∗)
prop
.
In
other
words,
(up
to
some
standard
general
nonsense)
one
may
think
of
(∗)
tech
as
the
concatenation
of
(∗)
prop
with
the
following
assertion:
(∗)
iner
Any
θ
:
Π
U
K
→
Π
Y
K
as
in
(∗)
tech
necessarily
factors
through
the
quotient
Π
U
K
→
Π
X
K
.
(Here,
“iner”
stands
for
“inertia”
–
since
the
kernel
of
Π
U
K
→
Π
X
K
is
generated
by
inertia
groups,
so
(∗)
iner
is
the
assertion
that
θ
:
Π
U
K
→
Π
Y
K
always
sends
inertia
groups
in
Π
U
K
to
the
identity.)
We
will
come
back
to
the
issue
of
(∗)
iner
later,
but
for
now,
let
us
concentrate
on
outlining
the
proof
of
(∗)
prop
,
since
it
is
this
which
is
the
core
of
the
paper.
Thus,
let
us
assume
that
we
have
been
given
a
continuous
surjective
homomorphism:
θ
:
Π
X
K
→
Π
Y
K
over
Γ
K
.
We
would
like
to
somehow
manufacture
out
of
θ
a
morphism
from
X
K
to
Y
K
.
A
natural,
naive
way
to
start
is
the
following:
First,
by
replacing
X
K
and
Y
K
by
appropriate
étale
coverings,
we
may
assume
that
neither
is
hyperelliptic
(see,
e.g.,
Lemma
10.4
(4)).
Let
us
write
D
X
(respectively,
D
Y
)
for
H
0
(X
K
,
ω
X
K
/K
)
(respectively,
H
0
(Y
K
,
ω
Y
K
/K
)),
the
space
of
global
differentials
on
X
K
(respectively,
Y
K
).
Then
it
is
well-known
that
X
K
def
(respectively,
Y
K
)
embeds
naturally
in
the
projective
space
P
X
=
P(D
X
)
(respectively,
def
P
Y
=
P(D
Y
)).
Moreover,
since
the
p-adic
étale
cohomology
of
X
K
or
Y
K
can
be
computed
as
the
group
cohomology
of
the
respective
pro-p
geometric
fundamental
group,
it
follows
that
the
surjection
θ
induces
a
Γ
K
-equivariant
injection
θ
H
:
H
1
(Y
K
,
Q
p
)
→
H
1
(X
K
,
Q
p
)
8
Moreover,
it
is
well-known
(see,
e.g.,
[Falt1],
[Tate])
that
the
Γ
K
-module
H
1
(X
K
,
Q
p
)
is
Hodge-Tate,
and,
moreover,
that
if
we
tensor
H
1
(X
K
,
Q
p
)
over
Q
p
with
C
p
(1)
(where
the
“(1)”
is
a
“Tate
twist”),
and
take
Γ
K
-invariants,
we
naturally
recover
the
space
D
X
of
differentials.
Thus,
since
θ
H
is
Γ
K
-equivariant,
if
we
tensor
θ
H
over
Q
p
with
C
p
(1)
and
take
Γ
K
-invariants,
we
obtain
from
θ
(in
a
natural
way)
a
K-linear
injection
θ
D
:
D
Y
→
D
X
and
hence
a
rational
map
from
P
X
to
P
Y
.
Thus,
in
some
sense,
without
doing
anything
terribly
new
(i.e.,
we
have
only
just
applied
results
known
to
Tate
since
the
1960’s),
we
have
already
come
relatively
close
to
constructing
a
morphism
from
X
K
to
Y
K
.
Indeed,
what
we
have
done
is
to
construct
a
morphism
θ
D
from
differentials
on
Y
K
to
differentials
on
X
K
,
which
we
would
like
to
hope
arises
as
the
pull-back
map
on
differentials
associated
to
a
morphism
from
X
K
to
Y
K
.
Moreover,
since
X
K
(respectively,
Y
K
)
is
canonically
embedded
in
P
X
(respectively,
P
Y
),
it
follows
from
elementary
algebraic
geometry
that
the
“only”
thing
we
need
to
show
is
⊗i
i
def
that
θ
D
preserves
relations:
That
is,
if
i
is
a
positive
integer,
let
D
X
=
H
0
(X
K
,
ω
X
).
K
/K
i
Similarly,
we
have
D
Y
.
Note
that
multiplication
of
differential
forms
defines
a
natural
morphism
i
D
Y
→
D
Y
i
Let
us
denote
the
kernel
of
this
morphism
by
R
i
.
Thus,
R
i
is
the
set
of
relations
(of
degree
i)
defining
Y
K
as
a
subvariety
of
P
Y
.
Moreover,
by
composing
the
i
th
tensor
power
of
θ
D
i
with
the
natural
morphism
from
the
i
th
tensor
power
of
D
X
to
D
X
,
we
obtain
a
morphism
i
κ
:
i
i
D
Y
→
D
X
Then
we
shall
say
that
θ
:
Π
X
K
→
Π
Y
K
preserves
relations
if
κ
i
(R
i
)
=
0
for
all
positive
integers
i.
As
stated
earlier,
once
we
know
that
θ
preserves
relations,
it
follows
from
elementary
algebraic
geometry
(see
[Harts],
Chapter
II)
that
the
rational
map
from
P
X
to
P
Y
defined
by
θ
D
induces
a
(dominant)
morphism
X
K
→
Y
K
,
as
desired.
(Once
one
has
this
morphism
X
K
→
Y
K
,
the
fact
that
the
map
that
it
induces
on
fundamental
groups
coincides
with
θ
is
a
matter
of
general
nonsense
–
for
details,
we
refer
to
the
argument
preceding
Theorem
14.1.)
Thus,
to
review
what
we
have
done
so
far,
we
have
reduced
the
main
problem
to
showing
that
any
θ
:
Π
X
K
→
Π
Y
K
as
in
(∗)
prop
preserves
relations.
It
should
be
emphasized
at
this
point,
that
so
far
everything
that
we
have
done
has
been
painless
general
nonsense
–
the
substantive
mathematical
core
of
the
argument
is
yet
to
come
(after
some
more
general
nonsense
in
the
next
few
paragraphs).
The
next
step
is
to
introduce
a
field
L,
9
as
follows:
Let
us
assume
for
simplicity
that
X
K
extends
to
a
stable
curve
X
over
O
K
.
Let
p
∈
X
be
an
irreducible
component
of
the
special
fiber
of
X
.
Then
the
completion
of
the
local
ring
O
X
,
p
is
a
p-adically
complete
discrete
valuation
ring
O
L
,
whose
quotient
field
we
denote
by
L.
The
space
Ω
L
of
p-adically
continuous
differentials
of
L
over
K
is
then
a
one-dimensional
L-vector
space.
Thus,
L
is,
in
some
sense,
one-dimensional
over
K.
Alternatively,
one
may
think
of
Spec(L)
roughly
as
some
sort
of
small
p-adic
open
set
in
X
K
.
At
any
rate,
from
the
construction
of
L,
it
follows
that
we
have
a
natural
L-valued
point
ξ
X
:
Spec(L)
→
X
K
.
Since
L
is
“one-dimensional,”
it
is
easy
to
check
that
the
restriction
map
on
differentials
i
D
X
→
Ω
⊗i
L
(where
“Ω
⊗i
L
”
denotes
the
tensor
product
of
i
copies
of
Ω
L
over
L)
is
injective.
Thus,
instead
of
checking
that
κ
i
(R
i
)
=
0,
it
suffices
to
check
that
the
composite
κ
iL
:
i
D
Y
→
Ω
⊗i
L
of
κ
i
with
the
above
restriction
map
vanishes
on
R
i
.
In
other
words,
we
would
like
to
compute
κ
iL
.
To
“compute”
κ
iL
,
we
need
to
go
back
to
looking
at
fundamental
groups.
First
of
all,
let
us
observe
that
by
functoriality
of
the
fundamental
group,
the
natural
L-valued
point
ξ
X
:
Spec(L)
→
X
K
defines
a
morphism
α
L
X
:
Γ
L
→
Π
X
K
whose
composite
with
the
augmentation
Π
X
K
→
Γ
K
is
the
morphism
Γ
L
→
Γ
K
on
Galois
groups
induced
by
the
inclusion
of
fields
K
⊆
L.
Moreover,
if
we
compose
α
L
X
with
θ,
we
obtain
a
morphism
α
L
Y
:
Γ
L
→
Π
Y
K
Now
let
us
suppose
that
we
know
that
α
L
Y
is
geometric,
i.e.,
that
it
arises
from
some
L-
valued
point
ξ
Y
:
Spec(L)
→
Y
K
of
Y
K
.
Then
it
follows
immediately
from
the
theory
of
[Falt1]
that
κ
iL
may
be
computed
as
the
restriction
map
on
differentials
associated
to
ξ
Y
.
But
it
is
clear
that
this
restriction
map
on
differentials
annihilates
R
i
.
Thus,
to
summarize:
in
order
to
show
that
θ
preserves
relations,
it
suffices
to
show
that
α
L
Y
is
geometric.
As
remarked
earlier,
the
above
prefatory
remarks
are
just
“general
nonsense.”
The
mathematical
core
of
the
present
paper
lies
in
showing
that
given
a
geometric
α
L
X
,
together
prop
L
,
the
resulting
α
Y
is
again
geometric.
The
proof
that
with
a
θ
:
Π
X
K
→
Π
Y
K
as
in
(∗)
L
α
Y
is
geometric
is
long
and
intricate,
and
can
be
divided
roughly
into
four
parts:
10
(1)
First,
we
consider
a
K-valued
point
x
∈
X
K
(K)
of
X
K
,
and
its
asso-
ciated
map
on
fundamental
groups
α
K
X
:
Γ
K
→
Π
X
K
.
It
is
elementary
to
show
that
the
process
of
passing
from
the
point
x
∈
X
K
(K)
to
the
arithmetic
first
Chern
class
of
its
associated
line
bundle
def
K
2
=
c
1
(O
X
K
(x))
∈
H
et
(X
K
,
Z
p
(1))
=
H
2
(Π
X
K
,
Z
p
(1))
η
X
can
be
carried
out
just
by
working
with
α
K
X
(see
the
first
half
of
Section
7
K
for
details
on
this
“group-theoretic
recipe”
for
concocting
η
X
out
of
α
K
X
).
K
K
On
the
other
hand,
by
composing
α
X
with
θ,
we
get
an
α
Y
:
Γ
K
→
Π
Y
K
.
The
first
step
then
is
to
show
that:
If
one
carries
out
this
recipe
for
α
K
Y
(which
may
or
may
not
arise
geometrically)
so
as
to
obtain
a
class
η
Y
K
∈
H
2
(Π
Y
K
,
Z
p
(1)),
then
η
Y
K
is
the
arithmetic
first
Chern
class
of
a
line
bundle
(necessarily
of
degree
prime
to
p)
on
Y
K
.
This
is
the
topic
of
Sections
1
through
5.
L
(2)
Next,
we
go
back
to
L-valued
points,
and
their
associated
α
L
X
,
α
Y
.
L
Then,
just
as
in
the
case
of
K-valued
points,
one
can
form
η
Y
,
and
one
would
like
to
know
that
η
Y
L
is
the
arithmetic
first
Chern
class
of
def
a
line
bundle
(necessarily
of
degree
prime
to
p)
on
Y
L
=
Y
K
⊗
K
L.
This
is
technically
much
more
intricate
than
the
K-valued
case,
and
is
done
(roughly
speaking)
by
thinking
about
the
“difference”
between
the
Chern
class
of
a
K-valued
point
and
of
an
L-valued
point,
and
showing
that
this
difference
has
special
properties
that
are
preserved
by
θ.
This
allows
one
to
derive
the
assertion
in
the
L-valued
case
from
the
corresponding
assertion
in
the
K-valued
case,
which
was
already
handled
in
(1)
above.
This
reduction
is
the
topic
of
Sections
6
and
7.
(3)
Recall
that
we
would
like
to
show
that
α
L
Y
is
geometric.
We
know
from
(2)
above
that
at
least
there
exists
a
line
bundle
on
Y
L
of
degree
prime
to
p.
Then
an
elementary
algebraic
geometry
argument
shows
that
this
implies
the
existence
of
a
rational
point
of
Y
L
defined
over
a
tamely
ramified
(this
will
be
crucial
in
Step
(4)!)
extension
L
of
L.
Although
this
portion
of
the
proof
is
technically
rather
trivial,
its
discovery
was
a
key
step
in
the
creation
of
the
proof
of
Theorem
A.
This
portion
of
the
proof
is
discussed
in
Section
8.
(4)
Finally,
by
applying
(3)
to
all
the
curves
in
a
certain
tower
of
coverings
of
the
original
Y
L
,
we
obtain
a
collection
of
rational
points
of
the
curves
of
this
tower
that
are
defined
over
tamely
ramified
extensions
of
L.
Then
by
using
Faltings’
p-adic
Hodge
theory
([Falt1]),
we
show
that
these
rational
points
necessarily
converge
p-adically
to
a
single
L-valued
point
of
Y
L
=
Y
K
⊗
K
L
whose
associated
Γ
L
→
Π
Y
K
is
necessarily
equal
to
11
L
α
L
Y
.
This
completes
the
proof
of
the
geometricity
of
α
Y
.
This
portion
of
the
proof
is
given
in
Sections
9
and
10
(and
applied
in
Section
13).
As
discussed
above,
once
one
knows
the
geometricity
of
α
L
Y
,
one
can
conclude
the
preser-
vation
of
relations
–
this
is
discussed
in
Sections
11
through
13.
Finally,
the
proof
of
(∗)
iner
(i.e.,
the
difference
between
(∗)
tech
and
(∗)
prop
)
is
given
in
Section
14.
Unfortunately,
in
order
to
show
(∗)
iner
,
it
is
necessary
to
go
through
all
the
steps
discussed
so
far
for
a
given
θ
:
Π
U
K
→
Π
Y
K
,
and
then
to
conclude
(∗)
iner
from
the
preservation
of
relations.
This
makes
the
proof
much
more
technically
intricate
than
it
would
be
if
one
could
prove
(∗)
iner
from
some
sort
of
a
priori
argument,
and
then
prove
preservation
of
relations
only
for
θ
:
Π
X
K
→
Π
Y
K
as
in
(∗)
prop
.
We
would
like
to
close
this
outline
of
the
proof
of
Theorem
A
by
discussing
Step
(1)
above
(i.e.,
the
content
of
Sections
1
through
5)
in
greater
detail.
The
reason
for
this
is
that
Step
(1)
is
what
allowed
us
to
generalize
the
“isomorphisms
only”
result
of
[Mzk2]
to
the
homomorphism
result
of
the
present
paper.
First
of
all,
let
us
recall
the
notion
of
the
Malčev
completion
(cf.,
e.g.,
[Del],
§9)
of
Δ
X
.
In
fact,
we
shall
only
need
the
“smallest
nontrivial
part
of
Malčev
completion”:
concretely,
a
unipotent
algebraic
group
over
Q
p
,
which
we
denote
by
M
X
,
whose
representations
are
the
same
as
continuous
representations
of
Δ
X
/[Δ
X
,
[Δ
X
,
Δ
X
]]
on
some
Q
p
-vector
space
V
which
is
equipped
with
a
Δ
X
-invariant
filtration
on
whose
subquotients
Δ
X
acts
trivially.
In
fact,
since
unipotent
algebraic
groups
are
equivalent
to
their
Lie
algebras,
we
shall
consider
instead
the
Lie
algebra
M
X
of
M
X
.
Next,
observe
that
any
section
α
K
X
:
Γ
K
→
Π
X
K
defines
(by
conjugation)
a
true
action
(i.e.,
not
just
an
action
up
to
inner
automorphisms)
of
Γ
K
on
Δ
X
,
hence
on
M
X
.
Relative
to
this
action,
their
exists
a
unique
“weight
zero
quotient”
M
X
⊗
Q
p
C
p
→
Z
X
.
Here,
we
call
the
quotient
“weight
zero”
because
it
is
the
maximal
quotient
M
X
⊗
Q
p
C
p
→
Q
for
which
the
action
of
Γ
K
on
Q
is
such
that
Q
has
a
filtration
by
Γ
K
-submodules
whose
subquotients
are
Γ
K
-equivariantly
isomorphic
to
C
p
.
Moreover,
this
quotient
M
X
⊗
Q
p
C
p
→
Z
X
is
independent
of
the
choice
of
section
α
K
X
.
Step
(1)
is
based
on
the
following
pair
of
observations:
(i)
On
the
one
hand,
if
α
K
X
arises
geometrically,
then
Z
X
is
“Hodge-
Tate.”
(By
abuse
of
terminology,
we
use
the
term
“Hodge-Tate”
here
to
mean
that
Z
X
has
a
C
p
-basis
of
Γ
K
-invariant
elements.)
This
is
the
content
of
Proposition
3.5.
K
is
the
(ii)
On
the
other
hand,
if
Z
X
is
“Hodge-Tate,”
then
the
class
η
X
first
Chern
class
of
a
line
bundle.
This
is
essentially
the
content
of
the
calculation
performed
in
Proposition
4.4
(see
also
Lemma
7.3).
Moreover,
if
one
starts
with
a
surjective
θ
:
Π
X
K
→
Π
Y
K
,
then
Z
X
maps
naturally
to,
and
in
fact,
surjects
onto
Z
Y
.
Thus,
if
one
starts
with
an
α
K
X
that
arises
geometrically,
then
one
knows
from
Observation
(i)
above
that
Z
X
is
Hodge-Tate,
but
any
quotient
of
a
Hodge-Tate
representation
of
Γ
K
–
i.e.,
such
as
Z
Y
–
is
always
Hodge-Tate,
so
Observation
(ii)
above
thus
allows
one
to
complete
Step
(1).
12
At
this
point,
the
reader
may
wonder
how
the
author
stumbled
upon
the
two
key
observations
of
the
preceding
paragraph.
In
fact,
the
author
first
realized
what
was
going
on
by
considering
the
ordinary
case
(see
Section
1
for
a
detailed
discussion).
In
this
case,
the
“weight
zero
quotient”
exists
at
the
level
of
groups,
without
passing
to
Malčev
completions
or
tensoring
with
C
p
:
namely,
(in
the
notation
of
Section
1)
it
is
the
quotient
et
Δ
X
→
Δ
et
X
(which
thus
gives
rise
to
a
quotient
Π
X
K
→
Π
X
K
).
Moreover,
as
is
shown
in
Section
1,
it
is
elementary
to
show
that
Δ
X
→
Δ
et
X
can
be
recovered
group-theoretically,
and
that
every
section
Γ
K
→
Π
X
K
that
arises
geometrically
(from
a
point
of
X
K
)
induces
et
a
fixed,
group-theoretically
constructible
section
θ
X
:
Γ
K
→
Π
et
X
K
of
Π
X
K
→
Γ
K
.
These
observations
are
enough
to
complete
Step
(1)
(in
the
ordinary
case).
Thus,
one
may
regard
the
discussion
in
the
preceding
paragraphs
(which
is
valid
in
the
nonordinary
case)
as
simply
the
result
of
generalizing
the
observations
discussed
in
this
paragraph
in
the
ordinary
case
to
the
possibly
nonordinary
case
by
means
of
the
technical
machinery
of
the
Malčev
completion
and
p-adic
Hodge
theory.
Finally,
let
us
make
the
following
observation:
As
one
can
see
from
the
key
argument
discussed
above,
in
fact,
really,
one
does
not
need
all
of
Δ
X
.
That
is
to
say,
in
the
above
argument
(concerning
the
issue
of
when
Z
X
is
Hodge-Tate),
one
actually
only
uses
the
quotient
Δ
X
/[Δ
X
,
[Δ
X
,
Δ
X
]].
It
is
this
observation
that
is
behind
the
truncated
generalizations
(Theorems
A
and
A
)
of
Theorem
A.
Acknowledgements:
I
would
like
to
thank
A.
Tamagawa:
(i)
for
inspiring
me
by
means
of
his
paper
[Tama];
(ii)
for
numerous
discussions
during
which
I
presented
the
proof
of
the
main
results
of
the
present
paper;
(iii)
for
advice
on
surmounting
two
technical
problems
(see
the
Remark
at
the
end
of
Section
12;
Lemmas
15.6
and
15.8);
(iv)
for
explaining
to
me
the
“general
nonsense
argument”
preceding
Theorem
14.1;
(v)
for
explaining
to
me
basic
facts
concerning
the
Malčev
completion,
which
were
of
immense
importance
in
developing
the
theory
discussed
in
this
paper.
Also,
I
would
like
to
thank
T.
Saito
and
G.
Faltings
for
pointing
out
various
minor
errors
in
earlier
versions
of
this
manuscript.
Section
0:
Preliminaries
and
Notations
Let
p
be
a
prime
number.
Throughout
this
paper
the
symbol
“∧”
over
or
to
the
upper-
right
of
an
object
will
denote
the
p-adic
completion
of
that
object.
Let
K
be
a
Q
p
-algebra.
Let
Ω
be
an
algebraically
closed
field.
Then
given
a
base-point
b
∈
Hom
Ring
(K,
Ω),
we
can
form
the
algebraic
fundamental
group
π
1
(Spec(K),
b)
Typically,
the
choice
of
base-point
b
will
not
be
important
for
us,
so
we
shall
write
Γ
K
for
π
1
(Spec(K),
b).
Suppose
that
Spec(K)
is
a
“universal
covering
space”
for
Spec(K)
such
13
that
b
arises
from
a
ring
homomorphism
K
→
Ω.
Then
we
may
think
of
Γ
K
as
Gal(K/K).
We
shall
denote
the
étale
cohomology
of
Spec(K)
by
H
i
(K,
−).
If
M
is
a
profinite
abelian
group
with
a
continuous
Γ
K
-action,
then
M
naturally
defines
an
inverse
system
{F
α
}
of
locally
constant
sheaves
on
the
étale
site
of
Spec(K),
and
we
shall
write
H
i
(K,
M)
for
the
inverse
limit
(over
α)
of
the
H
i
(K,
F
α
).
Note
that
for
each
r
∈
Z,
Z
p
(r)
(where
the
“(r)”
is
a
Tate
twist)
has
a
natural
structure
of
Γ
K
-module.
Definition
0.1.
We
shall
call
K
a
p-adic
field
if
it
is
the
quotient
field
of
a
p-adically
complete,
mixed
characteristic
discrete
valuation
ring
O
K
.
We
shall
denote
the
residue
field
(respectively,
maximal
ideal)
of
O
K
by
k
(respectively,
m
K
).
We
shall
call
K
a
p-adic
local
field
if
k
is
a
finite
field.
def
If
K
is
a
p-adic
field,
and
K
is
an
algebraic
closure
of
K,
then
K
→
Ω
=
K
determines
a
base-point
“b,”
and
we
have
Γ
K
=
Gal(K/K).
In
this
case,
H
i
(K,
−)
is
equal
to
the
continuous
group
cohomology
of
the
profinite
group
Γ
K
.
If
K
is
a
p-adic
local
field,
then
K
is,
in
fact,
a
finite
extension
of
Q
p
.
Now
(without
any
assumptions
on
the
Q
p
-algebra
K),
let
us
assume
that
we
are
given
a
hyperbolic
curve
X
K
→
Spec(K)
over
K
of
type
(g,
r).
(By
“curve,”
we
shall
always
mean
a
smooth,
one-dimensional,
geometrically
connected
scheme
over
the
base.
By
“type
(g,
r),”
we
mean
that
X
K
⊗
K
K
is
obtained
by
removing
r
mutually
nonintersecting
K-
valued
points
from
a
proper
curve
over
K
of
genus
g.
By
“hyperbolic,”
we
mean
that
2g
−
2
+
r
≥
1.)
When
X
K
is
proper,
we
shall
denote
its
Jacobian
(an
abelian
scheme
over
K)
by
J
X
(or
J
X
K
when
several
bases
are
in
use
and
it
is
necessary
to
specify
the
base
in
question).
Let
us
assume
that
X
K
is
equipped
with
a
base-point
x
∈
X
K
(Ω)
(which
is
compatible
def
def
with
the
base-point
b
of
Spec(K)).
Then
we
can
form
Π
prf
=
π
1
(X
K
,
x)
and
Δ
prf
X
X
=
π
1
(X
K
,
x).
(The
use
of
“Δ”
to
denote
the
geometric
fundamental
group
may
be
new
to
some
readers.
Here,
we
use
“Δ”
partly
because
“Π”
is
already
used
for
the
arithmetic
fundamental
group
and
partly
to
conform
to
the
notations
of
[Falt1],
a
reference
on
which
the
present
paper
depends
heavily.)
Occasionally,
to
avoid
confusion,
we
shall
also
use
the
prf
prf
notation
Δ
prf
X
K
for
Δ
X
.
Let
Δ
X
be
the
maximal
pro-p
quotient
of
Δ
X
.
Since
the
kernel
prf
prf
of
Δ
prf
X
→
Δ
X
is
normal
in
Π
X
K
,
we
may
form
the
quotient
of
Π
X
K
by
this
kernel,
and
call
the
resulting
quotient
group
Π
X
K
.
Thus,
we
have
an
exact
sequence
1
→
Δ
X
→
Π
X
K
→
Γ
K
→
1
Moreover,
this
exact
sequence
induces
a
representation
14
def
ρ
X
:
Γ
K
→
Out(Δ
X
)
=
Aut(Δ
X
)/Inn(Δ
X
)
into
the
outer
automorphism
group
of
Δ
X
.
(Here,
“Aut(Δ
X
)”
(respectively,
“Inn(Δ
X
)”)
denotes
the
group
of
continuous
automorphisms
(respectively,
inner
automorphisms)
of
Δ
X
.)
Conversely,
it
is
well-known
(see,
e.g.,
[Tama],
§7,
A.)
that
the
above
exact
sequence
can
be
recovered
from
ρ
X
.
Next,
we
would
like
to
introduce
some
terminology
particular
to
pro-p
groups.
Let
Δ
be
a
pro-p
group
(i.e.,
a
topological
group
obtained
by
taking
an
inverse
limit
of
finite
groups
of
p-power
order).
Let
Δ
⊆
Δ
be
the
unique
normal
subgroup
of
Δ
with
the
following
property:
Δ
→
Δ/Δ
is
the
maximal
(topologically)
Hausdorff
abelian
quotient
of
Δ
which
def
def
is
annihilated
by
p.
For
i
≥
0,
let
Δ
<0>
=
Δ;
Δ
<i+1>
=
(Δ
<i>
)
.
Thus,
we
obtain
a
descending
series
of
closed
normal
(even
characteristic!)
subgroups
.
.
.
⊆
Δ
<i>
⊆
.
.
.
⊆
Δ.
Note
that
since
Δ,
being
a
pro-p
group,
is
“pro-solvable,”
it
follows
that
the
intersection
of
all
the
Δ
<i>
is
{1}.
Moreover,
if
Δ
is
topologically
finitely
generated,
it
follows
that
the
Δ/Δ
<i>
are
all
finite
groups.
Definition
0.2.
We
shall
refer
to
any
one
of
the
Δ
<i>
as
a
p-derivate
of
Δ.
Thus,
in
particular,
if
Δ
=
Δ
X
,
then
it
follows
(by
the
structure
of
the
fundamental
group
of
an
algebraic
curve
in
characteristic
zero)
that
Δ
is
topologically
finitely
generated,
so
the
Δ/Δ
<i>
are
all
finite
groups.
Next,
let
us
consider
the
Kummer
sequence
on
X
K
,
i.e.,
the
exact
sequence
of
étale
sheaves
on
X
K
given
by
0
→
Z/p
n
Z(1)
→
G
m
→
G
m
→
0
(for
n
≥
1).
(Here,
the
“(1)”
is
a
“Tate
twist,”
and
the
morphism
from
G
m
to
G
m
is
given
by
raising
to
the
(p
n
)
th
power.)
The
connecting
morphism
induced
on
étale
cohomology
by
the
Kummer
sequence
then
gives
us
a
morphism
H
1
(X
K
,
G
m
)
→
H
2
(X
K
,
(Z/p
n
Z)(1)).
Now
suppose
that
L
is
a
line
bundle
on
X
K
.
Applying
the
connecting
morphism
just
considered
to
L
(which
defines
an
element
of
H
1
(X
K
,
G
m
)),
we
obtain
a
compatible
system
of
classes
in
H
2
(X
K
,
(Z/p
n
Z)(1))
(for
each
n
≥
1),
hence
a
class
c
1
(L)
∈
H
2
(X
K
,
Z
p
(1)).
Definition
0.3.
We
shall
refer
to
c
1
(L)
as
the
arithmetic
first
Chern
class
of
L.
Finally,
we
have
the
following
elementary
technical
result,
which
states
that
the
étale
cohomology
of
a
hyperbolic
curve
may
be
computed
as
the
group
cohomology
of
its
fun-
damental
group:
Lemma
0.4.
Assume
that
K
is
a
field.
For
all
integers
i,
r,
the
natural
morphisms
H
i
(Δ
X
,
Z
p
(r))
→
H
i
(X
K
,
Z
p
(r));
H
i
(Δ
X
K
×
K
X
K
,
Z
p
(r))
→
H
i
(X
K
×
K
X
K
,
Z
p
(r))
15
and
H
i
(Π
X
K
,
Z
p
(r))
→
H
i
(X
K
,
Z
p
(r));
H
i
(Π
X
K
×
K
X
K
,
Z
p
(r))
→
H
i
(X
K
×
K
X
K
,
Z
p
(r))
are
isomorphisms.
Proof.
By
the
Leray-Serre
spectral
sequence,
it
suffices
to
prove
that
the
morphisms
of
the
first
line
are
isomorphisms.
Let
Z
be
X
K
or
X
K
×
K
X
K
.
Then
it
follows
by
general
nonsense
that
it
suffices
to
check
that
for
any
finite
étale
Galois
covering
Y
→
Z
of
p-power
order,
and
any
cohomology
class
η
∈
H
i
(Y,
F
p
)
(where
i
>
0),
there
exists
a
finite
étale
covering
Y
→
Y
of
p-power
order
such
that
η|
Y
=
0.
By
the
Künneth
formula,
it
suffices
to
do
the
case
Z
=
X
K
.
Then
what
we
must
check
is
trivial
for
i
>
2
(since
then
η
=
0
to
begin
with),
and
clear
for
i
=
1
(by
the
relationship
between
étale
coverings
and
H
1
).
If
i
=
2,
then
it
suffices
to
take
Y
→
Y
such
that
Y
→
Y
has
degree
p
over
every
connected
component
of
Y
.
This
completes
the
proof.
Section
1:
The
Ordinary
Case
Let
p
be
a
prime
number.
Let
K
be
a
p-adic
field
with
algebraically
closed
residue
field.
Then,
as
discussed
in
Section
0,
the
absolute
Galois
group
of
K
will
be
denoted
Γ
K
.
Let
X
K
→
Spec(K)
be
a
hyperbolic
curve
over
K
of
type
(g,
r).
In
this
Section,
let
us
also
assume
that
X
K
admits
a
stable
extension
X
→
Spec(O
K
)
over
O
K
.
By
this,
we
mean
that
there
exists
a
(necessarily
unique)
r-pointed
stable
(in
particular,
proper)
curve
X
→
Spec(O
K
)
of
genus
g
such
that
X
is
the
complement
in
X
of
the
images
of
the
r
marking
sections
of
X.
Let
us
write
X
k
(respectively,
X
k
)
for
X
⊗
O
K
k
(respectively,
X
⊗
O
K
k).
Remark.
Recall
that
an
r-pointed
stable
curve
of
genus
g
(where
2g
−
2
+
r
≥
1)
is
a
proper,
flat
morphism
f
:
C
→
S,
together
with
r
mutually
disjoint
sections
σ
1
,
.
.
.
,
σ
r
:
S
→
C,
such
that
(1)
The
geometric
fibers
of
f
are
connected,
reduced,
of
arithmetic
genus
g,
and
have
at
most
nodes
as
singularities.
(2)
The
sheaf
ω
C/S
(σ
1
+
.
.
.
+
σ
r
)
(i.e.,
the
sheaf
of
sections
of
the
dualizing
bundle
of
C
over
S
with
poles
of
order
≤
1
at
the
divisors
defined
by
the
images
of
the
sections
σ
1
,
.
.
.
,
σ
r
)
is
relatively
ample
over
S.
We
refer
to
[DM],
[Knud]
for
more
details.
16
Now
we
make
the
following
Definition
1.1.
We
shall
call
X
K
ordinary
if
the
Jacobian
of
every
connected
component
of
the
normalization
of
the
curve
X
k
is
an
ordinary
abelian
variety.
In
this
Section,
we
would
like
to
assume
that
X
K
is
ordinary.
Under
this
assumption,
it
is
well-known
that
π
1
(X
k
)
(p)
(where
the
“(p)”
denotes
the
maximal
pro-p
quotient)
is
a
free
2
pro-p
group
of
rank
g.
(Indeed,
this
follows
from
the
fact
that
H
et
(X
k
,
F
p
)
=
0
(which
may
be
shown
by
using
the
long
exact
sequence
in
étale
cohomology
obtained
by
considering
F
p
as
the
kernel
of
“1−
Frobenius”
acting
on
O
X
k
),
plus
Proposition
2.3
of
Chapter
III,
§3,
of
[Sha].)
Moreover,
since
étale
coverings
of
X
k
lift
uniquely
to
characteristic
zero,
it
follows
that
we
have
a
continuous
surjection:
(p)
X
:
Π
X
K
→
π
1
(X
k
)
If
we
restrict
X
to
Δ
X
,
we
obtain
a
surjection
Δ
X
→
π
1
(X
k
)
(p)
.
In
the
following,
we
shall
regard
π
1
(X
k
)
(p)
as
a
quotient
of
Δ
X
via
this
surjection.
Let
us
denote
this
quotient
et
by
Δ
et
X
.
Note
that
the
kernel
of
the
surjection
Δ
X
→
Δ
X
is
normal
as
a
subgroup
of
Π
X
K
.
Thus,
by
taking
the
quotient
of
Π
X
K
by
this
kernel,
we
obtain
a
quotient
Π
X
K
→
Π
et
X
K
.
In
other
words,
we
have
an
exact
sequence
et
1
→
Δ
et
X
→
Π
X
K
→
Γ
K
→
1
together
with
a
surjection
et
ζ
X
:
Π
et
X
K
→
Δ
X
which
is
the
identity
on
Δ
et
X
.
Now
observe
that
the
kernel
of
ζ
X
projects
isomorphically
to
Γ
K
.
Thus,
we
obtain
a
section
θ
X
:
Γ
K
→
Π
et
X
K
Next,
let
us
observe
that
every
element
of
Im(θ
X
)
commutes
with
every
element
of
Δ
et
X
.
Indeed,
this
follows
from
the
fact
that
such
commutation
relations
hold
after
projection
et
by
Π
et
X
K
→
Γ
K
,
plus
the
fact
that
Im(θ
X
)
=
Ker(ζ
X
)
is
normal
in
Π
X
K
(and
maps
isomorphically
to
Γ
K
via
the
projection
Π
et
X
K
→
Γ
K
).
On
the
other
hand,
since,
as
is
well-known
(see,
e.g.,
[Tama],
§1,
Propositions
1.1,
1.11),
pro-p
free
groups
(of
rank
≥
2)
have
trivial
centers,
it
thus
follows
that
we
obtain
the
following
“group-theoretic”
characterization
of
θ
X
(when
g
≥
2):
17
et
Lemma
1.2.
Suppose
that
g
≥
2.
Then
the
section
θ
X
:
Γ
K
→
Π
et
X
K
of
Π
X
K
→
Γ
K
is
et
the
unique
section
whose
image
commutes
with
every
element
of
Δ
et
X
⊆
Π
X
K
.
On
the
other
hand,
the
quotient
Δ
X
→
Δ
et
X
(and
hence
also
the
quotient
Π
X
K
→
et
Π
X
K
)
can
also
be
reconstructed
group-theoretically
by
means
of
the
following
condition
on
an
open
normal
subgroup
H
⊆
Δ
X
:
(∗)
et
Let
N
⊆
Δ
X
be
any
subgroup
such
that
H
⊆
N
and
N/H
is
cyclic.
def
Then
there
exists
a
surjection
N
ab
=
N/[N,
N
]
→
Q,
where
Q
is
a
free
Z
p
-module
of
rank
one,
with
the
following
properties:
(i)
there
exists
an
open
subgroup
Γ
⊆
Γ
K
that
stabilizes
N
and
N
ab
→
Q
and
acts
trivially
on
Q;
(ii)
the
surjection
N
→
N/H
factors
through
Q.
Then
we
have
the
following
“group-theoretic”
characterization
of
the
quotient
Δ
X
→
Δ
et
X
:
Lemma
1.3.
The
kernel
of
Δ
X
→
Δ
et
X
is
the
intersection
of
all
open
normal
subgroups
H
⊆
Δ
X
such
that
H
satisfies
the
condition
(∗)
et
.
Proof.
The
proof
is
entirely
the
same
as
that
of
Sections
3
and
8
of
[Mzk1].
The
basic
idea
is
that
if
the
covering
corresponding
to
Δ
X
/H
is
not
étale
(over
O
K
),
then
it
has
nontrivial
inertia
subgroups.
Such
inertia
subgroups
have
nontrivial
cyclic
subgroups.
If
we
then
apply
(∗)
et
to
the
case
where
N/H
is
one
of
these
nontrivial
cyclic
subgroups,
then
we
have
a
contradiction,
since
the
quotient
N
→
N/H
factors
through
the
quotient
N
→
Q;
moreover,
the
quotient
N
→
Q
necessarily
corresponds
to
a
covering
which
is
étale
over
O
K
because
of
the
assumption
concerning
the
action
of
Γ
⊆
Γ
K
on
Q.
Let
us
review
what
we
have
done
so
far.
So
far,
we
have:
et
(1)
constructed
quotients
Π
X
K
→
Π
et
X
K
and
Δ
X
→
Δ
X
,
as
well
as
a
section
Γ
K
→
Π
et
X
K
via
various
geometric
considerations
concerning
X
K
;
(2)
shown
(when
g
≥
2)
that
the
above
quotients
and
section
may
be
reconstructed
entirely
“group-theoretically.”
Here,
we
pause
to
make
the
following
Remark
Concerning
the
Term
“Group-Theoretic.”
In
[Mzk1]
and
[Mzk2],
we
imparted
mathematical
rigor
to
the
term
“group-theoretic”
(cf.
the
remark
on
this
issue
in
Section
14
of
[Mzk2])
by
specifying
that
it
meant
“preserved
by
isomorphism.”
In
the
present
paper,
however,
we
would
like
to
consider
homomorphisms
which
are
not
necessarily
iso-
morphisms.
Thus,
in
the
present
paper,
when
we
wish
to
state
that
a
certain
property
or
object
is
preserved
by
such
homomorphisms,
we
shall
state
this
explicitly
without
using
18
the
term
“group-theoretic.”
In
the
above
discussion,
however,
we
recommend
the
reader
to
simply
accept
this
term
at
the
level
of
“common
sense,”
since
we
will
not
use
the
“group-
theoreticity”
stated
in
any
of
the
results
of
this
Section
in
the
proof
of
any
of
the
main
theorems
of
this
paper.
Finally,
before
continuing,
we
make
the
following
important
observation:
Let
α
x
:
Γ
K
→
Π
X
K
denote
the
section
of
Π
X
K
→
Γ
K
determined
(up
to
composition
with
an
inner
automorphism
induced
by
an
element
of
Δ
X
)
by
a
point
x
∈
X
K
(K).
Denote
by
et
et
α
et
x
:
Γ
K
→
Π
X
K
the
composite
of
α
x
with
Π
X
K
→
Π
X
K
.
Lemma
1.4.
We
have
α
et
x
=
θ
X
.
et
Proof.
It
suffices
to
show
that
the
composite
of
α
et
x
with
the
surjection
ζ
X
:
Π
X
K
→
Δ
et
X
is
trivial.
Interpreted
geometrically,
this
simply
means
that
“the
pull-back
of
any
étale
covering
Y
→
X
(i.e.,
étale
over
O
K
)
to
Spec(K)
via
x
is
the
trivial
étale
covering
of
Spec(K).”
But
this
assertion
follows
immediately
from
the
fact
that
O
K
is
strictly
henselian.
Section
2:
Review
of
Galois
Cohomology
Let
K
be
a
p-adic
field
whose
residue
field
is
perfect.
Let
S
→
Spec(O
K
)
be
a
geometrically
connected
smooth
morphism,
and
let
D
⊆
S
be
a
relative
(over
O
K
)
divisor
with
normal
crossings.
Let
us
write
S
log
for
the
log
scheme
obtained
by
equipping
S
with
the
log
structure
defined
by
D
(as
in
[Kato]).
Let
us
also
assume
that
S
is
small
(in
the
sense
of
[Falt1],
[Falt2]):
Recall
that
this
simply
means
that
S
is
affine,
say,
equal
to
Spec(R),
and,
moreover,
étale
over
some
O
K
[X
1
,
.
.
.
,
X
d
]
in
such
a
way
that
D
is
schematically
the
inverse
image
of
the
zero
locus
of
the
function
X
1
·
.
.
.
·
X
d
.
The
reason
we
wish
to
deal
with
small
(S,
D)
is
that
in
[Falt1],
certain
Galois
cohomology
groups
associated
to
such
(S,
D)
are
computed
explicitly
by
means
of
the
theory
of
almost
étale
extensions.
The
purpose
of
this
Section
is
to
review
certain
consequences
of
the
theory
of
[Falt1]
and
[Falt2]
that
are
of
relevance
to
us
in
this
paper.
def
Let
R
K
=
R
⊗
O
K
K.
Let
R
K
→
R
K
be
the
maximal
extension
of
R
K
which
is
étale
(respectively,
R)
be
the
outside
of
D
.
Let
R
be
the
normalization
of
R
in
R
.
Let
R
K
K
def
def
p-adic
completion
of
R
(respectively,
R);
let
R
K
=
R
⊗
O
K
K,
R
K
=
R
⊗
O
K
K.
Let
R
O
K
def
O
)
⊗
O
K.
Let
Γ
R
def
be
the
p-adic
completion
of
R
⊗
O
O
;
R
=
(
R
=
Gal(R
K
/R
K
).
K
K
K
K
K
Thus,
we
have
a
natural
surjection
Γ
R
→
Γ
K
whose
kernel
we
denote
by
Δ
prf
R
.
This
gives
us
a
natural
exact
sequence:
1
→
Δ
prf
R
→
Γ
R
→
Γ
K
→
1
19
Now
we
would
like
to
compute
some
Galois
cohomology
groups:
Lemma
2.1.
For
all
j
∈
Z,
we
have:
(1)
H
0
(Δ
prf
R
,
R
K
(j))
=
R
K
(j);
(2)
H
1
(Δ
prf
R
,
R
K
(j))
=
Ω
S
log
/O
K
⊗
R
R
K
(j
−
1).
(Here
the
“(j)”
is
a
Tate
twist.)
Proof.
This
follows
from
Theorem
4.4
of
Section
I
of
[Falt1].
Indeed,
(1)
follows
from
[Falt1],
Section
I,
Theorem
4.4,
(i)
(applied
to
the
case
i
=
0,
and
tensored
with
K(j)),
while
(2)
follows
from
[Falt1],
Section
I,
Theorem
4.4,
(iv)
(tensored
with
K(j)).
)=
R
K
;
(ii)
H
n
(Γ
K
,
R
(j))
=
0
(for
j
=
0;
Lemma
2.2.
We
have:
(i)
H
0
(Γ
K
,
R
K
K
)=
R
K
.
n
=
0,
1);
(iii)
H
1
(Γ
K
,
R
K
Proof.
This
result
is
due
to
[Tate]
in
the
case
R
=
O
K
.
(Indeed,
in
this
case,
(i)
and
(iii)
follow
from
[Tate],
§3.3,
Theorem
1;
while
(ii)
–
at
least
in
the
case
j
=
1
(the
proof
for
arbitrary
j
=
0
is
entirely
similar)
–
follows
from
[Tate],
§3.3,
Theorem
2.)
The
slightly
more
general
result
stated
here
(i.e.,
for
R
not
necessarily
equal
to
O
K
)
follows
by
the
same
argument
as
that
employed
by
Tate
in
the
case
R
=
O
K
(the
point
being
that
in
general,
R
is
O
K
-flat).
Alternatively,
the
cohomology
groups
in
the
Lemma
can
also
be
computed
using
almost
étale
extensions
as
in
[Falt1].
Lemma
2.3.
)=
R
(−1))
=
0.
K
;
(ii)
H
n
(Γ
R
,
R
For
n
=
0,
1,
we
have:
(i)
H
n
(Γ
R
,
R
K
K
Proof.
This
follows
from
the
Leray-Serre
spectral
sequence
(applied
to
the
exact
sequence
of
groups
that
appears
directly
before
Lemma
2.1),
plus
the
preceding
two
Lemmas.
Now
let
us
assume
that
we
are
given
an
r-pointed
stable
log-curve
f
log
:
X
log
→
S
log
of
genus
g,
where
2g
−
2
+
r
≥
1.
(By
this
we
mean
that
X
log
is
obtained
by
pulling
log
log
back
the
universal
log-curve
C
log
→
M
g,r
via
some
log
morphism
S
log
→
M
g,r
.
The
log
structure
of
the
universal
log
curve
is
defined
by
the
divisor
with
normal
crossings
which
is
the
union
of
the
marked
points
and
the
singular
fibers.
See,
e.g.,
[Mzk3],
Section
3,
for
more
details;
cf.
also
the
Remark
preceding
Definition
1.1
of
this
paper
for
a
review
of
def
the
notion
of
a
“pointed
stable
curve.”)
Let
U
=
S
−
D.
Then
taking
the
relative
first
cohomology
module
of
f|
U
K
in
the
étale
topology
with
coefficients
in
Z
p
gives
rise
to
a
local
system
over
U
K
,
hence
a
Γ
R
-module
H
∨
.
As
a
Z
p
-module,
H
∨
is
free
of
rank
=
2g
20
(if
r
=
0)
and
=
2g
+
r
−
1
(if
r
>
0).
Let
H
be
the
Γ
R
-module
given
by
Hom
Z
p
(H
∨
,
Z
p
).
Now
let
def
H
0
=
(R
1
f
∗
O
X
)
∨
(a
vector
bundle
over
S,
or
alternatively,
a
projective
R-module,
of
rank
g)
and
def
H
1
=
(f
∗
ω
X
log
/S
log
)
∨
(a
vector
bundle
over
S,
or
alternatively,
a
projective
R-module,
whose
rank
is
=
g
if
r
=
0
and
=
g
+
r
−
1
if
r
>
0).
Proposition
2.4.
There
is
a
natural
exact
sequence
(1)
→
H
⊗
R
→H
⊗
R
0
→
H
1
⊗
R
R
K
Z
p
K
0
R
K
→
0
which
is
compatible
with
the
natural
action
of
Γ
R
on
all
three
terms.
Proof.
This
is
an
immediate
consequence
of
the
“Comparison
Theorem”
(Theorem
6.2
of
[Falt2]).
The
validity
of
the
proof
of
this
Theorem
given
in
[Falt2]
has
been
disputed
by
various
mathematicians.
However,
since
Proposition
2.4
is
a
relatively
weak
consequence
of
the
“Comparison
Theorem,”
it
already
follows
from
the
portion
of
[Falt2]
that
is
not
in
dispute.
(That
is
to
say,
we
need
only
that
M
is
“Hodge-Tate,”
not
that
it
is
“crystalline.”)
Alternatively,
although
the
sort
of
parametrized
(i.e.,
over
a
base
S)
Hodge-Tate
decom-
position
that
we
need
here
is
not
stated
in
[Falt1],
it
follows
immediately
from
the
theory
of
[Falt1]
by
exactly
the
same
proof
as
that
of
the
main
result
of
[Falt1].
Yet
another
proof
of
this
sort
of
result
is
given
in
[Hyodo]
(the
final
Theorem
–
i.e.,
the
“relative
version”
–
in
[Hyodo],
§0.3),
although
here
we
need
the
(relatively
straightforward)
logarithmic
generalization
of
[Hyodo]’s
result.
Section
3:
The
Weight
Zero
Quotient
In
this
Section,
we
maintain
the
notation
of
the
preceding
Section.
The
purpose
of
this
Section
is
to
give
a
(rather
weak)
nonabelian
analogue
of
Proposition
2.4.
In
fact,
a
much
stronger
“nonabelian
comparison
theorem”
for
the
fundamental
group
of
a
curve
can
be
proven,
but
since
I
do
not
know
of
any
place
where
such
a
result
has
been
written
up,
and,
moreover,
in
this
paper,
only
a
relatively
weak
nonabelian
comparison
theorem
is
needed,
I
decided
instead
to
give
a
rather
short
ad
hoc
treatment
of
this
issue
which
will,
nonetheless,
be
sufficient
for
the
purposes
of
this
paper.
21
log
Let
us
denote
the
fundamental
group
of
X
K
by
Π
prf
X
K
.
Thus,
we
have
a
natural
prf
prf
surjection
Π
X
K
→
Γ
R
.
As
usual,
we
denote
the
kernel
of
this
surjection
by
Δ
prf
X
⊆
Π
X
K
,
and
the
maximal
pro-p
quotient
of
Δ
prf
X
by
Δ
X
.
Moreover,
by
forming
the
quotient
of
prf
prf
Π
X
K
by
the
kernel
of
Δ
X
→
Δ
X
,
we
obtain
Π
X
K
.
Thus,
just
as
in
Section
1,
we
have
an
exact
sequence
1
→
Δ
X
→
Π
X
K
→
Γ
R
→
1
Moreover,
any
section
σ
:
S
→
X
of
X
→
S
whose
image
avoids
the
marked
points
and
nodes
defines
a
section
π
σ
:
Γ
R
→
Π
X
K
of
the
above
exact
sequence.
Such
a
section
π
σ
then
defines
an
action
of
Γ
R
on
Δ
X
by
conjugation.
Note
that
until
one
specifies
the
section
α
σ
:
Γ
R
→
Π
X
K
,
one
only
has
an
outer
action
of
Γ
R
on
Δ
X
(i.e.,
an
action
defined
only
up
to
inner
automorphisms);
that
is
to
say,
in
general,
there
is
no
natural
action
(in
the
usual,
non-outer
sense)
of
Γ
R
on
Δ
X
until
one
specifies
a
section
of
Π
X
K
→
Γ
R
.
The
next
step
is
to
introduce
the
Malčev
completion
of
Δ
X
.
We
refer
to
[Del],
[NT]
for
more
details.
In
fact,
for
our
purposes,
it
will
be
sufficient
to
consider
a
truncated
form
of
the
Malčev
completion
of
Δ
X
.
This
truncated
form
may
be
defined
as
follows:
Let
C
be
the
category
of
finite
dimensional
Q
p
-vector
spaces
V
equipped
with
a
continuous
Δ
X
-action
that
factors
through
Δ
X
/[Δ
X
,
[Δ
X
,
Δ
X
]]
and
which
admits
a
Δ
X
-invariant
filtration
on
whose
subquotients
Δ
X
acts
trivially.
(The
morphisms
of
this
category
are
the
Q
p
[Δ
X
]-
linear
morphisms
V
→
V
.)
Then
C
is
a
Tannakian
category
over
Q
p
,
hence
gives
rise
to
an
algebraic
group
M
X
over
Q
p
.
Moreover,
this
algebraic
group
M
X
is
unipotent,
hence
corresponds
to
a
nilpotent
Lie
algebra
M
X
.
For
any
Q
p
-algebra
A,
we
shall
write
(M
X
)
A
(respectively,
(M
X
)
A
)
for
M
X
⊗
Q
p
A
(respectively,
M
X
⊗
Q
p
A).
Let
us
write
M
X
[1]
for
the
commutator
[M
X
,
M
X
]
of
this
Lie
algebra,
and
M
X
[0]
for
def
the
quotient
M
X
/M
X
[1].
Then
M
X
[0]
may
be
identified
with
H
Q
p
=
H
⊗
Z
p
Q
p
(where
H
is
as
in
the
discussion
following
Lemma
2.3
in
Section
2).
Thus,
we
get
an
exact
sequence
0
→
M
X
[1]
→
M
X
→
M
X
[0]
=
H
Q
p
→
0
Moreover,
the
commutator
[−,
−]
defines
a
surjection
c
X
:
∧
2
H
Q
p
→
M
X
[1]
Now
observe
that
although
there
is
no
natural
action
of
Γ
R
on
M
X
(unless
one
chooses
a
section
of
Π
X
K
→
Γ
R
),
there
is
nonetheless
a
natural
action
of
Γ
R
on
M
X
[0]
and
M
X
[1]
with
respect
to
which
c
X
is
equivariant.
Now
we
have
the
following
classical
result
Lemma
3.1.
The
kernel
of
c
X
is
zero
if
r
>
0.
If
r
=
0,
then
the
kernel
of
c
X
is
one-
∨
dimensional,
and
equal
to
the
image
of
the
dual
to
the
intersection
form
∧
2
H
Q
→
Q
p
(−1)
p
(defined
by
the
cup
product
pairing
on
the
cohomology
of
Δ
X
).
22
Proof.
If
r
>
0,
then
Δ
X
is
a
free
pro-p
group.
Thus,
in
this
case
the
result
follows
by
consideration
of
the
fact
that
a
representation
on
a
vector
space
V
of
the
free
group
on
n
generators
is
the
same
as
giving
n
endomorphisms
of
V
.
If
r
=
0,
then
Δ
X
is
the
quotient
of
a
free
pro-p
group
by
a
single
relation;
it
is
this
relation
which
generates
the
kernel
of
c
X
.
Moreover,
it
is
easy
to
see
from
the
well-known
explicit
form
of
this
relation
that
its
image
in
∧
2
H
Q
p
is
precisely
as
specified
in
the
statement
of
the
Lemma
(cf.
the
discussion
in
§2
of
[NT]).
Next,
we
would
like
to
construct
a
certain
special
quotient
Z
X
of
(M
X
)
.
By
the
R
K
well-known
categorical
equivalence
between
unipotent
algebraic
groups
and
nilpotent
Lie
algebras
(cf.
[Del],
§9),
this
quotient
will
define
a
quotient
Z
X
of
(M
X
)
.
The
construc-
R
K
tion
of
Z
X
from
(M
X
)
consists
of
two
steps.
The
first
step
is
as
follows:
Consider
the
R
K
surjection
→
∧
2
H
⊗
R
∧
2
H
⊗
Z
p
R
K
0
R
K
defined
by
projecting
by
means
of
the
surjection
in
the
short
exact
sequence
of
Proposition
2.4.
It
follows
from
Lemmas
2.3
(ii)
(in
the
case
n
=
0)
and
3.1
that
this
surjection
factors
through
(M
X
[1])
.
Thus,
we
obtain
a
surjection
R
K
(M
X
[1])
R
K
→
∧
2
H
0
⊗
R
R
K
By
pushing
forward
the
exact
sequence
0
→
M
X
[1]
→
M
X
→
M
X
[0]
→
0
(tensored
over
)
via
this
surjection,
we
thus
obtain
a
Lie
algebra
U
(over
R
).
Thus,
we
Q
with
R
p
K
X
have
a
surjection
of
Lie
algebras
(M
X
)
R
K
K
→
U
X
,
together
with
an
exact
sequence
0
→
U
X
[1]
→
U
X
→
U
X
[0]
→
0
def
,
and
U
[0]
=
(M
[0])
.
where
U
X
[1]
=
[U
X
,
U
X
]
=
∧
2
H
0
⊗
R
R
K
X
X
R
K
Now
we
come
to
the
second
step
in
the
construction
of
Z
X
.
First
let
us
denote
by
def
the
surjection
defined
by
the
surjection
in
the
short
exact
U
X
[0]
→
U
X
[0,
0]
=
H
0
⊗
R
R
K
sequence
of
Proposition
2.4.
Let
U
X
[0,
1]
⊆
U
X
[0]
be
the
kernel
of
this
surjection.
Let
B
X
⊆
U
X
denote
the
inverse
image
of
U
X
[0,
1]
⊆
U
X
[0]
under
the
surjection
U
X
→
U
X
[0].
Then
it
follows
from
the
definition
of
U
X
that
B
X
is
an
abelian
Lie
algebra
which,
in
fact,
lies
in
the
center
of
the
Lie
algebra
of
U
X
.
(Indeed,
this
will
follow
as
soon
as
we
show
the
vanishing
of
[B
X
,
U
X
],
which
is
equal
to
the
image
of
U
X
[0,
1]
∧
U
X
[0]
⊆
∧
2
H
Q
p
under
the
;
composite
of
c
:
∧
2
H
→
M
[1]
with
the
projection
M
[1]
→
U
[1]
=
∧
2
H
⊗
R
X
Q
p
X
X
X
0
R
K
but
this
image
is
zero,
by
the
definition
of
U
X
[0,
1].)
This
observation
implies,
in
particular,
that
although,
a
priori,
we
have
only
an
outer
action
of
Γ
R
on
U
X
,
hence
on
B
X
,
in
fact,
we
get
a
natural
(non-outer)
action
of
Γ
R
on
B
X
.
(That
is,
the
point
is
that
the
various
23
actions
of
Γ
R
on
B
X
arising
from
different
sections
of
Π
X
K
→
Γ
R
differ
only
by
an
automorphism
on
B
X
induced
by
conjugation
by
some
element
of
Δ
X
;
but
since
B
X
lies
in
the
center
of
U
X
,
it
follows
that
such
an
automorphism
of
B
X
is
always
equal
to
the
identity.)
Next,
let
us
note
that
we
have
exact
sequences
0
→
B
X
→
U
X
→
U
X
[0,
0]
→
0
and
0
→
U
X
[1]
→
B
X
→
U
X
[0,
1]
→
0
where
the
latter
exact
sequence
is
an
exact
sequence
of
Γ
R
-modules
(a
statement
which
has
meaning
as
a
result
of
the
observation
at
the
end
of
the
preceding
paragraph).
Now
observe
that
Lemma
2.3
(ii)
(for
n
=
0,
1
–
note
that
here
we
use
that
U
[1]
=
∧
2
H
⊗
R
X
0
R
K
(1)
is
“of
weight
one”)
implies
that
the
is
“of
weight
zero,”
while
U
X
[0,
1]
=
H
1
⊗
R
R
K
latter
exact
sequence
admits
a
unique
Γ
R
-equivariant
splitting:
B
X
→
U
X
[1].
Moreover,
since
B
X
is
(as
observed
above)
contained
in
the
center
of
the
Lie
algebra
U
X
,
it
follows
that
the
kernel
of
this
splitting
B
X
→
U
X
[1]
forms
a
Lie
ideal
in
U
X
.
Thus,
if
we
then
push
forward
the
former
exact
sequence
via
this
surjection
B
X
→
U
X
[1],
we
obtain
a
Lie
algebra
Z
X
.
As
usual,
this
Lie
algebra
fits
into
an
exact
sequence
0
→
Z
X
[1]
→
Z
X
→
Z
X
[0]
→
0
Moreover,
one
has
natural
identifications:
Z
X
[1]
=
∧
2
Z
X
[0]
(via
the
commutator
map);
(induced
by
the
surjection
of
the
short
exact
sequence
of
Propo-
and
Z
[0]
=
H
⊗
R
X
0
R
K
.
Finally,
as
noted
above,
Z
defines
sition
2.4).
In
particular,
Z
X
[0]
is
of
rank
g
over
R
K
X
a
unipotent
algebraic
group
Z
X
.
Definition
3.2.
We
shall
refer
to
Z
X
as
the
weight
zero
quotient
of
Δ
X
(even
though
it
is
not
literally
a
quotient).
(Here,
the
“Z”
of
Z
X
stands
for
the
“zero”
of
“weight
zero
quotient.”)
Now
let
us
fix
a
(continuous)
section
α
:
Γ
R
→
Π
X
K
Then
α
induces
an
action
of
Γ
R
on
Δ
X
.
Since
Z
X
was
formed
naturally
–
and,
for
that
matter,
group-theoretically
–
from
Δ
X
,
we
thus
obtain
an
action
of
Γ
R
on
Z
X
which,
in
24
general,
will
depend
on
the
choice
of
α.
The
next
issue
we
would
like
to
consider
is
to
what
extent
the
sequence
of
Γ
R
-modules
0
→
Z
X
[1]
→
Z
X
→
Z
X
[0]
→
0
admits
a
Γ
R
-equivariant
section.
At
any
rate,
this
exact
sequence
defines
an
extension
class
η
α
∈
H
1
(Γ
R
,
(Z
X
[0])
∨
⊗
Z
X
[1])
R
K
and
Z
[1]
=
By
Lemma
2.3
(i)
(for
n
=
1
–
here
we
use
that
both
Z
X
[0]
=
H
0
⊗
R
R
K
X
∧
2
Z
X
[0]
are
“of
weight
zero”),
it
follows
that
this
Galois
cohomology
group
may
be
identified
naturally
with
2
(H
∨
0
⊗
R
∧
H
0
)
⊗
R
R
K
K
.
That
is
to
say,
one
may
think
of
η
α
as
a
section
of
a
certain
vector
bundle
over
R
Proposition
3.3.
Suppose
that
α
arises
as
the
α
σ
associated
to
some
section
σ
:
S
→
X
(whose
image
avoids
the
marked
points
and
nodes
–
cf.
the
discussion
at
the
beginning
of
this
Section).
Then
η
α
=
0.
Proof.
Since
everything
is
functorial,
one
reduces
immediately
to
the
universal
case,
as
follows:
In
the
present
context,
the
essential
data
that
we
begin
with
is
a
r-pointed
stable
curve
of
genus
g
(i.e.,
f
log
:
X
log
→
S
log
),
plus
a
section
(i.e.,
σ).
The
moduli
stack
for
this
data
is
(a
certain
dense
open
substack
of)
the
tautological
curve
C
→
M
g,r
over
the
moduli
stack
M
g,r
of
r-pointed
stable
curves
of
genus
g
over
Z
p
.
Thus,
by
“restriction,”
it
suffices
to
prove
the
Proposition
in
the
case
where
S
is
étale
over
the
algebraic
stack
C.
Also,
since
everything
involved
commutes
with
base-extension,
it
is
easy
to
see
that
we
may
assume
that
k
is
algebraically
closed.
Then
over
some
dense
open
T
⊆
(S
−
D)
⊗
O
K
k,
the
Jacobian
of
X
⊗
O
K
k
will
be
ordinary.
Thus,
every
point
of
β
∈
S(O
K
)
that
maps
Spec(k)
⊆
Spec(O
K
)
into
T
defines
an
ordinary
(in
the
sense
of
Definition
1.1)
hyperbolic
curve
Y
β
→
Spec(K)
over
K
(by
restricting
(the
complement
of
the
marking
sections
in)
X
→
S
to
β).
Next,
observe
that
that
if
we
restrict
the
section
σ
to
the
point
β,
we
get
a
section
et
σ
β
:
Spec(K)
→
Y
β
which
induces
an
action
of
Γ
K
on
Δ
et
Y
β
(i.e.,
the
quotient
Δ
Y
β
→
Δ
Y
β
considered
in
Section
1).
By
Lemmas
1.2
and
1.4,
this
action
is,
in
fact,
the
trivial
action.
Now
I
claim
that
in
the
case
of
such
an
ordinary
curve:
The
quotient
Δ
Y
β
→
Δ
et
Y
β
(considered
in
Section
1)
induces
a
natural
(in
particular,
Γ
K
-equivariant)
isomorphism
of
the
weight
zero
quotient
25
Z
Y
β
of
Δ
Y
β
(=
Δ
X
)
with
the
Malčev
completion
of
Δ
et
Y
β
(truncated
at
the
second
step
and
tensored
over
Z
with
K).
p
Indeed,
it
follows
immediately
from
the
construction
of
Z
Y
β
that
the
surjection
Δ
Y
β
→
Δ
et
Y
β
induces
a
morphism
from
Z
Y
β
to
the
“weight
zero
quotient”
(i.e.,
constructed
in
a
fashion
analogous
to
the
construction
of
Z
Y
β
)
of
Δ
et
Y
β
.
On
the
other
hand,
since
the
action
of
et
Γ
K
on
Δ
Y
β
is
trivial,
it
follows
that
the
“weight
zero
quotient”
of
Δ
et
Y
β
is
just
the
Malčev
Moreover,
completion
of
Δ
et
(truncated
at
the
second
step
and
tensored
over
Z
with
K).
p
Y
β
it
is
clear
that
the
map
induced
on
abelianizations
by
this
morphism
from
Z
Y
β
to
this
weight
zero
quotient
of
Δ
et
Y
β
is
an
isomorphism.
(Indeed,
surjectivity
follows
from
the
fact
that
we
are
dealing
(by
definition)
with
various
quotients
of
a
single
object;
injectivity
then
follows
from
Proposition
2.4
and
Lemma
2.3
(ii)
(for
n
=
0),
and
the
fact
that
both
Thus,
(since
Δ
et
is
a
free
pro-p
group
of
rank
g
abelianizations
are
of
rank
g
over
K.)
Y
β
–
cf.
the
discussion
following
Definition
1.1)
we
conclude
that
we
get
an
isomorphism
as
stated
in
the
“claim.”
This
completes
the
verification
of
the
claim.
Thus,
any
minimal
choice
of
generators
of
Δ
et
Y
β
(which
will
necessarily
be
fixed
by
Γ
K
)
defines
a
splitting
of
the
sequence
of
Γ
K
-modules
0
→
Z
X
[1]
→
Z
X
→
Z
X
[0]
→
0
Here
the
Γ
K
-action
is
given
by
composing
the
Γ
R
-action
considered
above
with
the
mor-
phism
–
well-defined
up
to
composition
with
an
inner
automorphism
(which
does
not
bother
us
since
this
inner
automorphism
corresponds
to
a
coboundary
in
the
computation
of
the
cohomology
class
η
α
that
we
are
interested
in)
–
Γ
K
→
Γ
R
defined
by
β.
But
from
the
definition
of
the
extension
class
η
α
,
this
means
that
the
restriction
2
η
α
|
β
∈
(H
∨
0
⊗
R
∧
H
0
)
⊗
R,β
K
of
η
α
to
the
point
β
is
zero.
On
the
other
hand,
if
η
α
is
zero
when
restricted
to
any
such
β
(i.e.,
any
β
∈
S(O
K
)
that
maps
Spec(k)
⊆
Spec(O
K
)
into
T
),
it
is
clear
that
η
α
itself
must
be
zero.
This
completes
the
proof
of
the
Proposition.
Note
that
so
far,
in
this
Section
and
the
last,
we
have
been
dealing
with
families
of
curves,
parametrized
over
a
base
S.
Before
continuing
on
to
the
next
Section,
it
is
worthwhile
to
go
back
to
the
case
of
“a
single
curve”
over
K
in
order
to
make
explicit
the
consequences
for
such
single
curves
of
the
theory
developed
thus
far.
Thus,
let
X
K
→
Spec(K)
be
a
hyperbolic
curve.
As
in
Section
1,
we
have
an
exact
sequence
1
→
Δ
X
→
Π
X
K
→
Γ
K
→
1
26
Then
let
us
first
note
that
the
theory
reviewed
in
Section
2
can
be
applied
in
the
present
nonparametrized
situation
as
well
to
produce
(following
precisely
the
same
recipe
as
in
the
parametrized
situation
considered
as
above)
a
Lie
algebra
Z
X
,
together
with
a
group
Z
X
(both
over
K).
Definition
3.4.
We
shall
refer
to
Z
X
as
the
weight
zero
quotient
of
Δ
X
.
Let
x
∈
X
K
(K).
Then
x
defines
a
section
α
x
:
Γ
K
→
Π
X
K
(well-defined
up
to
composition
with
an
inner
automorphism
arising
from
Δ
X
).
Moreover,
α
x
defines
an
action
of
Γ
K
on
Z
X
.
Proposition
3.5.
of
Γ
K
-modules
Relative
to
the
action
of
Γ
K
on
Z
X
defined
by
α
x
,
the
exact
sequence
0
→
Z
X
[1]
→
Z
X
→
Z
X
[0]
→
0
splits.
Proof.
Note
that
this
Proposition
is
not,
strictly
speaking,
a
special
case
of
Proposition
3.3
(i.e.,
where
we
take
S
=
O
K
),
since
Proposition
3.3
only
addresses
the
case
where
the
divisor
of
bad
reduction
is
flat
over
O
K
.
However,
by
means
of
specialization,
one
can
reduce
the
present
Proposition
to
the
“universal
case”
considered
during
the
proof
of
Proposition
3.3.
Moreover,
in
this
case,
Proposition
3.3
already
tells
us
that
the
relevant
exact
sequence
is
split.
This
completes
the
proof.
Remark.
Let
us
write
(Δ
X
)
Q
p
for
the
full
(i.e.,
not
truncated
as
above)
Malčev
completion
(cf.
[Del],
[NT])
of
Δ
X
.
Then
in
some
sense,
Proposition
3.5
above
is
a
truncated
version
of
a
theorem
that
states
“relative
to
the
action
of
Γ
K
on
(Δ
X
)
Q
p
defined
by
α
x
,
(Δ
X
)
Q
p
is
Hodge-Tate.”
Here,
since
(Δ
X
)
Q
p
is
an
inverse
limit
of
unipotent
algebraic
groups,
one
can
interpret
“Hodge-Tate”
to
mean
that
the
Lie
algebras
of
each
of
these
unipotent
algebraic
groups
are
Hodge-Tate
representations.
In
fact,
it
is
not
difficult
to
prove
that
(Δ
X
)
Q
p
is
Hodge-Tate
(even
for
arbitrary
higher-dimensional
smooth
X
K
→
Spec(K))
as
follows:
One
reduces
the
higher-dimensional
case
to
the
curve
case
by
cutting
with
hyperplane
sections.
Then
for
curves,
by
considering
the
universal
case,
one
can
reduce
to
the
case
of
curves
smooth
over
O
K
.
Finally,
for
curves
smooth
over
O
K
,
one
can
apply
the
techniques
of
[Falt1]
(by
considering
cohomology
spaces
with
coefficients
valued
in
unipotent
algebraic
groups
over
Q
p
).
This
shows
that
(Δ
X
)
Q
p
is
Hodge-Tate.
Moreover,
one
can
also
construct
∞
a
nontruncated
“weight
zero
quotient
(Δ
X
)
Q
p
→
Z
X
”
(that
is
to
say,
if
X
K
is
a
curve
of
∞
genus
g,
then
Z
will
be
(Γ
-equivariantly)
isomorphic
to
the
Malčev
completion
over
K
X
K
of
the
free
group
on
g
generators).
In
fact,
it
is
even
possible
to
show
that
(Δ
X
)
Q
p
is
de
Rham,
but
we
shall
not
pursue
such
issues
here
since
they
are
not
relevant
to
the
proof
of
27
the
main
results
of
this
paper.
Nevertheless,
we
remark
that
this
sort
of
theorem
has
been
verified
by
A.
Shiho
([Shiho]),
in
a
manuscript
in
preparation.
Section
4:
J-Geometric
Sections
We
maintain
the
notations
of
the
latter
portion
of
Section
3.
Moreover,
we
assume
that
our
curve
X
K
→
Spec(K)
is
proper
(hence
of
genus
≥
2)
and
that
X
K
(K)
=
∅.
In
this
Section,
we
would
like
to
consider
a
continuous
homomorphism
α
:
Γ
K
→
Π
X
K
which
defines
a
section
of
the
surjection
Π
X
K
→
Γ
K
.
The
section
α
defines
an
action
of
Γ
K
on
Δ
X
by
conjugation,
and
hence
also
an
action
of
Γ
K
on
Z
X
.
In
particular,
we
would
like
to
consider
the
significance
of
the
following
condition
on
α:
(∗)
spl
The
exact
sequence
of
Γ
K
-modules
(relative
to
the
action
defined
by
α)
0
→
Z
X
[1]
→
Z
X
→
Z
X
[0]
→
0
splits.
The
main
result
of
this
Section
is
to
show
that
the
group-theoretic
condition
(∗)
spl
is
equivalent
to
the
statement
that
α
is
“J
-geometric”
(a
term
which
means
that
α
acts
in
some
respects
as
if
it
came
from
a
geometric
point
x
∈
X
K
(K)
–
see
Definition
4.3
for
a
precise
definition).
To
do
this,
first
we
need
to
recall
certain
facts
concerning
Jacobians
and
their
funda-
(d)
mental
groups.
For
d
∈
Z,
let
J
X
→
Spec(K)
be
the
Picard
scheme
of
line
bundles
on
X
K
(d)
(0)
(d)
of
degree
d.
Thus,
J
X
is
a
torsor
over
J
X
=
J
X
,
the
Jacobian
of
X
K
.
Note
that
J
X
is
defined
even
if
X
K
does
not
admit
any
K-rational
points,
as
the
scheme
representing
the
étale
sheafification
of
the
usual
Picard
functor
of
degree
d
line
bundles.
Note
that
a
base-
point
x
∈
X
K
(Ω)
(where
Ω
is
the
algebraically
closed
field
of
Section
0)
may
be
regarded
as
a
degree
one
divisor
on
X
Ω
=
X
K
⊗
K
Ω,
hence
(by
multiplying
this
divisor
by
d)
we
(d)
obtain
a
point
x
d
∈
J
X
(Ω).
This
allows
us
to
define
the
arithmetic
fundamental
group
(d)
(d)
π
1
(J
X
,
x
d
),
as
well
as
its
geometric
counterpart
π
1
(J
X
,
x
d
).
If
we
form
the
quotient
(d)
K
(d)
of
π
1
(J
X
,
x
d
)
by
the
kernel
of
π
1
(J
X
,
x
d
)
→
Δ
(d)
(i.e.,
the
projection
to
the
maximal
J
X
K
pro-p
quotient
of
the
geometric
fundamental
group),
then
we
obtain
a
topological
group
Π
(d)
.
Moreover,
we
have
a
natural
exact
sequence
J
X
28
1
→
Δ
J
(d)
→
Π
J
(d)
→
Γ
K
→
1
X
X
(1)
Finally,
the
natural
embedding
X
K
→
J
X
induces
a
natural
surjection
Π
X
K
→
Π
(1)
,
J
X
whose
kernel
is
the
commutator
subgroup
of
Δ
X
.
Now
let
us
observe
that
we
can
reconstruct
Π
(d)
from
Π
X
K
→
Γ
K
,
as
follows.
First
of
J
X
all,
we
can
reconstruct
Π
(1)
as
the
quotient
of
Π
X
K
by
the
commutator
subgroup
of
Δ
X
.
J
X
(d)
Next,
let
us
observe
that,
since
(for
all
d
∈
Z)
J
X
is
a
J
X
-torsor,
considering
the
action
(d)
of
J
X
on
J
X
allows
one
to
identify
Δ
(d)
with
Δ
J
X
.
Thus,
we
have
an
exact
sequence
J
X
1
→
Δ
J
X
→
Π
J
(1)
→
Γ
K
→
1
X
(1)
(d)
If
we
consider
(for
nonzero
d),
the
morphism
J
X
→
J
X
given
by
multiplication
by
d,
we
d
see
that
the
result
of
pushing
forward
this
sequence
by
means
of
the
morphism
Δ
J
X
→
Δ
J
X
(i.e.,
multiplication
by
d)
gives
rise
to
an
exact
sequence
which
can
be
naturally
identified
with
the
exact
sequence
1
→
Δ
J
(d)
=
Δ
J
X
→
Π
J
(d)
→
Γ
K
→
1
X
X
This
completes
our
review
of
Jacobians
and
their
fundamental
groups.
Now
recall
from
the
definition
of
Z
X
[0]
in
Section
3
that
Z
X
[0]
may
be
identified
with
In
particular,
one
has
a
natural
Γ
-equivariant
the
weight
zero
portion
of
Δ
⊗
K.
J
X
Z
p
K
morphism
Δ
J
X
→
Z
X
[0].
Thus,
by
pushing
forward
the
exact
sequence
of
the
preceding
paragraph
(for
d
=
1)
by
means
of
this
morphism,
we
obtain
a
morphism
of
exact
sequences
of
topological
groups
as
follows:
Δ
J
X
⏐
⏐
−→
1
−→
Z
X
[0]
−→
1
−→
Π
(1)
J
X
⏐
⏐
−→
Γ
K
⏐
⏐
−→
Π
Z
(1)
−→
Γ
K
−→
J
X
1
1
1
Next,
we
would
like
to
consider
sections
of
Π
(1)
→
Γ
K
and
Π
Z
(1)
→
Γ
K
.
Recall
that
the
J
X
J
X
difference
between
any
two
sections
of
Π
(1)
→
Γ
K
(respectively,
Π
Z
(1)
→
Γ
K
)
is
given
by
J
X
J
X
an
element
of
H
1
(K,
Δ
J
X
)
(respectively,
H
1
(K,
Z
X
[0])).
Moreover,
if
θ
L
,
θ
M
:
Γ
→
Π
(1)
(1)
are
sections
that
arise
from
geometric
points
L,
M
∈
J
X
(K),
then
the
difference
def
δ
=
θ
L
−
θ
M
∈
H
1
(K,
Δ
J
X
)
29
J
X
maps
to
zero
in
H
1
(K,
Z
X
[0])
([BK],
Example
3.11
–
cf.
the
proof
of
Lemma
4.1
for
some
more
details).
Thus,
it
follows
that
the
section
θ
L
[0]
:
Γ
K
→
Π
Z
(1)
obtained
from
θ
L
J
X
by
composing
θ
L
with
Π
(1)
→
Π
Z
(1)
is
independent
(up
to
composition
with
an
inner
J
X
J
X
automorphism
of
Π
Z
(1)
arising
from
Z
X
[0]
⊆
Π
Z
(1)
)
of
the
choice
of
L.
In
particular,
as
J
X
J
X
(1)
long
as
J
X
(K)
is
nonempty
(which
is
the
case
here,
since
X
K
(K)
has
been
assumed
to
be
nonempty),
we
thus
obtain
a
canonical
section
(well-defined
up
to
composition
with
an
inner
automorphism
of
Π
Z
(1)
arising
from
Z
X
[0]
⊆
Π
Z
(1)
)
J
X
J
X
θ
geom
:
Γ
K
→
Π
Z
(1)
J
X
of
Π
Z
(1)
→
Γ
K
.
J
X
Now
let
θ
:
Γ
K
→
Π
(1)
be
an
arbitrary
section,
and
let
θ[0]
:
Γ
K
→
Π
Z
(1)
denote
J
X
J
X
the
section
induced
by
θ.
Then
taking
the
difference
between
θ[0]
and
θ
geom
defines
a
cohomology
class
δ
θ
∈
H
1
(K,
Z
X
[0])
=
H
0
(where
the
last
equality
follows
from
Lemma
2.3,
applied
in
the
case
R
K
=
K).
Now
we
have
the
following
result:
Lemma
4.1.
Suppose
that
the
residue
field
k
is
finite.
Then
there
exists
a
geometric
(1)
point
L
∈
J
X
(K)
such
that
θ
=
θ
L
if
and
only
if
δ
θ
=
0.
Proof.
This
follows
immediately
from
the
theory
of
[BK],
§3,
especially
Example
3.11,
plus
the
following
observation:
If
H
is
(as
in
Section
3)
the
Γ
K
-module
that
arises
as
the
abelianization
of
Δ
X
,
then
+
Ker{H
1
(K,
H)
→
H
1
(K,
H
⊗
B
DR
)}
(i.e.,
by
Lemma
3.8.1
of
[BK],
“H
g
1
”
in
the
notation
of
[BK])
is
equal
to
=
H
1
(K,
Z
[0])}
Ker{H
1
(K,
H)
→
H
1
(K,
H
⊗
K)
X
(where
the
last
equality
follows
from
Proposition
2.4
and
Lemma
2.2
(ii)
(for
n
=
1,
j
=
1)).
+
Indeed,
this
equality
of
kernels
follows
by
using
the
natural
filtration
on
B
DR
(whose
i
≥
0),
together
with
the
fact
that
H
1
(K,
H
⊗
K(i))
=0
subquotients
are
equal
to
K(i),
for
i
>
0
(by
Proposition
2.4,
Lemma
2.2
(ii)
(for
n
=
1,
j
>
0)).
30
Now,
we
return
to
considering
the
section
α
:
Γ
K
→
Π
X
K
.
Let
β
:
Γ
K
→
Π
X
K
be
any
section
that
arises
from
some
geometric
point
x
∈
X
K
(K).
We
would
like
to
compare
α
and
β,
and,
in
particular,
show
that,
under
the
assumption
(∗)
spl
,
α
and
β
are,
in
some
sense
(to
be
specified
precisely
below),
relatively
close.
Let
ζ
:
Γ
K
→
Δ
X
be
the
continuous
function
(not
necessarily
a
group
homomorphism)
such
that
α(γ)
=
ζ(γ)
·
β(γ)
∈
Π
X
K
for
all
γ
∈
Γ
K
.
(respectively,
If
γ
∈
Γ
K
,
then
let
us
write
γ
α
(φ)
∈
Z
X
(
K)
Now
let
φ
∈
Z
X
(
K).
for
the
result
of
letting
γ
act
on
φ
by
means
of
the
action
defined
by
α
γ
(φ)
∈
Z
(
K))
β
X
(respectively,
β).
Note
that
it
follows
from
the
construction
of
Section
3
that
we
have
a
For
∈
Δ
,
let
us
denote
by
the
image
of
natural
morphism
Δ
→
Z
(
K).
∈
Z
(
K)
X
X
Then
we
have
the
following:
in
Z
X
(
K).
Lemma
4.2.
X
Z
X
We
have:
γ
α
(φ)
=
ζ(γ)
Z
·
γ
β
(φ)
·
ζ(γ)
−1
Z
.
Proof.
Indeed,
this
follows
immediately
from
the
fact
that
the
respective
actions
of
Γ
K
are
induced
by
conjugation
by
α(γ)
and
β(γ)
inside
Π
.
on
Z
X
(
K)
X
K
Let
α
:
Γ
K
→
Π
X
K
be
any
section
of
Π
X
K
→
Γ
K
,
and
let
α
Z
:
Γ
K
→
Π
Z
(1)
be
the
section
obtained
by
composing
α
J
X
Z
with
Π
X
K
→
Π
(1)
.
Then
we
make
the
following
J
X
Definition
4.3.
We
shall
call
α
J
-geometric
if
α
Z
coincides
with
θ
geom
(up
to
compo-
sition
with
an
inner
automorphism
of
Π
Z
(1)
arising
from
Z
X
[0]
⊆
Π
Z
(1)
).
J
X
J
X
The
following
result
is
the
main
technical
observation
that
made
it
possible
to
substantially
strengthen
the
result
of
[Mzk2].
Proposition
4.4.
Let
α
:
Γ
K
→
Π
X
K
be
a
continuous
group
homomorphism
that
defines
a
section
of
Π
X
K
→
Γ
K
.
Then
α
satisfies
(∗)
spl
if
and
only
if
α
is
J
-geometric.
be
β-invariant,
i.e.,
invariant
under
the
action
of
Γ
on
Z
(
K)
Proof.
Let
φ
∈
Z
X
(
K)
K
X
defined
by
β.
We
would
like
to
calculate
the
action
of
Γ
K
on
φ
that
is
induced
by
α.
Thus,
for
γ
∈
Γ
K
,
we
have,
by
Lemma
4.2:
γ
α
(φ)
=
ζ(γ)
Z
·
γ
β
(φ)
·
ζ(γ)
−1
Z
=
ζ(γ)
Z
·
φ
·
ζ(γ)
−1
Z
31
Thus,
γ
α
(φ)
·
φ
−1
=
[ζ(γ)
Z
,
φ]
(where
the
brackets
denote
the
“commutator”).
Note
that
=
Z
[1]
=
∧
2
Z
[0]
=
(∧
2
H
)
⊗
K
[ζ(γ)
Z
,
φ]
∈
Z
X
[1](
K)
0
K
X
X
Now
it
remains
to
reinterpret
the
calculation
just
performed
in
terms
of
cohomology
classes.
Recall
the
class
2
2
η
α
∈
H
∨
0
⊗
K
∧
H
0
=
Hom
K
(H
0
,
∧
H
0
)
in
Z
[0](
K)
=
Z
[0].
Since
discussed
in
Section
3.
Let
φ[0]
be
the
image
of
φ
∈
Z
X
(
K)
X
X
φ
is
β-invariant,
it
follows
that
φ[0]
is
a
Γ
K
-invariant
element
of
Z
X
[0],
hence
belongs
to
=
Z
[0].
Let
δ
def
=
α
−θ
∈
H
1
(K,
Z
[0])
=
H
.
Then
it
is
immediate
H
⊆H
⊗
K
0
0
K
X
α
Z
geom
0
X
from
the
definitions
that
the
calculation
of
the
preceding
paragraph,
interpreted
in
terms
of
cohomology
classes,
becomes:
η
α
(φ[0])
=
δ
α
∧
(φ[0])
(Note
that
since
θ
geom
may
be
computed
–
cf.
the
discussion
preceding
Lemma
4.1
–
using
the
geometric
section
β,
it
follows
that
δ
α
is
precisely
the
cohomology
class
defined
by
the
in
Z
[0](
K)
=
Z
[0].)
Next
observe
that
this
image
of
the
cocycle
ζ(−)
Z
:
Γ
K
→
Z
X
(
K)
X
X
=
Z
[0]
=
equation
holds
for
all
β-invariant
φ,
and
that
every
element
of
H
⊆
H
⊗
K
0
0
K
X
lifts
to
a
β-invariant
φ
∈
Z
(
K).
Z
X
[0](
K)
(Indeed,
this
follows
from
Proposition
3.5
X
since
β
arises
from
a
point
of
X
K
(K).)
Thus,
it
follows
that
η
α
:
H
0
→
H
0
∧
H
0
is
simply
the
map
“δ
α
∧.”
In
particular,
η
α
=
0
if
and
only
if
δ
α
=
0.
This
completes
the
proof
of
the
Proposition.
Remark.
So
far
here
we
have
been
dealing
with
the
truncated
weight
zero
quotient
Δ
X
→
Z
X
,
but
it
would
be
interesting
also
to
see
what
happens
in
the
case
of
the
full
nontruncated
∞
weight
zero
quotient
Δ
X
→
Z
X
(as
in
the
Remark
at
the
end
of
Section
3).
For
instance,
∞
∞
if
the
action
of
α
on
Z
X
is
such
that
Z
X
has
“enough
invariants”
(i.e.,
there
exists
a
pro-
is
Γ
-equivariantly
isomorphic
to
Z
∞
),
does
algebraic
group
G
over
K
such
that
G
⊗
K
K
K
X
it
follow
that
α
itself
automatically
comes
from
a
geometric
point
x
∈
X
K
(K)?
Although
such
questions
are
beyond
the
scope
of
this
paper,
it
is
the
opinion
of
the
author
that
such
questions
deserve
further
study.
Note
that
this
sort
of
issue
is
closely
related
to
the
so-called
Section
Conjecture
–
cf.
the
Remark
following
Theorem
19.1
for
more
details
on
this
conjecture.
32
Section
5:
The
J-Geometricity
of
K-Valued
Points
Let
K
be
a
p-adic
field
with
perfect
residue
field.
Let
X
K
and
Y
K
be
proper
hyperbolic
curves
over
K.
Let
U
K
be
the
K-scheme
obtained
by
localizing
X
K
at
its
generic
point.
Thus,
the
underlying
topological
space
of
U
K
consists
of
one
point,
and
the
ring
of
functions
on
U
K
is
the
function
field
K
X
of
X
K
.
Note
in
particular
that
we
can
consider
Π
U
K
,
Π
Y
K
.
In
particular,
Π
U
K
is
a
certain
quotient
of
the
absolute
Galois
group
of
K
X
.
Let
us
assume
that
we
are
given
a
continuous
open
homomorphism
θ
:
Π
U
K
→
Π
Y
K
over
Γ
K
.
In
this
Section,
we
would
like
to
begin
the
proof
of
the
main
theorem
of
this
paper
by
showing
that
any
such
θ
necessarily
“maps
geometric
sections
to
J
-geometric
sections.”
(Naturally,
we
will
explain
below
precisely
what
is
meant
by
the
expression
in
quotes.)
First
observe
that
if
x
∈
X
K
(K)
is
any
K-valued
point,
then
we
can
form
the
com-
pletion
(K
X
)
x
of
the
field
K
X
with
respect
to
the
valuation
defined
by
x.
Moreover,
we
have
a
natural
morphism
Π
(K
X
)
x
→
Π
U
K
(well-defined
up
to
composition
with
conjugation
by
an
element
of
Δ
U
)
whose
image
is
“the”
(more
rigorously:
any
of
the
various
conju-
gate)
decomposition
group
associated
to
x.
Moreover,
as
is
well-known
(see,
e.g.,
[Ser2]),
(K
X
)
x
is
(noncanonically)
isomorphic
to
K((t))
(where
t
is
an
indeterminate),
so
Δ
(K
X
)
x
may
be
identified
with
Z
p
(1).
In
particular,
by
forming,
relative
to
some
isomorphism
(K
X
)
x
∼
=
K((t)),
the
field
extension
of
(K
X
)
x
corresponding
to
adjoining
a
compatible
system
of
p-power
roots
of
t
to
K((t)),
one
sees
immediately
that
Π
(K
X
)
x
→
Γ
K
admits
many
sections.
Definition
5.1.
We
shall
refer
to
as
geometric
any
section
Γ
K
→
Π
U
K
of
Π
U
K
→
Γ
K
obtained
by
composing
a
section
of
Π
(K
X
)
x
→
Γ
K
with
(any
one
of
the
conjugate
natural
homomorphisms)
Π
(K
X
)
x
→
Π
U
K
.
Note,
in
particular,
that
Π
(K
X
)
x
→
Π
X
K
factors
through
Γ
K
,
so
in
particular,
the
com-
posite
α
X
:
Γ
K
→
Π
X
K
with
Π
U
K
→
Π
X
K
of
any
geometric
section
α
U
:
Γ
K
→
Π
U
K
is
induced
by
some
point
x
:
Spec(K)
→
X
K
.
Let
α
U
:
Γ
K
→
Π
U
K
be
a
geometric
section.
By
composing
α
U
with
θ,
we
obtain
a
section
α
Y
:
Γ
K
→
Π
Y
K
.
In
this
Section,
we
would
like
to
prove
that
α
Y
is
necessarily
J
-geometric
in
the
sense
of
Definition
4.3.
To
do
this,
let
us
first
observe
that
we
have
a
diagram
of
continuous
morphisms
Δ
U
−→
⏐
⏐
θ|
Δ
U
Z
X
(
K)
−→
Z
Y
(
K)
Δ
Y
33
which
are
compatible
with
the
Γ
K
-actions
defined
by
α
U
and
α
Y
on
the
all
the
groups
involved.
(Here,
the
upper
horizontal
morphism
is
obtained
by
composing
Δ
U
→
Δ
X
with
Δ
→
Z
(
K).)
Then
we
have
the
following
X
X
→
Z
(
K)
Lemma
5.2.
There
exists
a
(natural)
surjective
Γ
K
-homomorphism
Z
X
(
K)
Y
that
makes
the
above
diagram
commute.
Proof.
Observe
that
the
kernel
of
Δ
U
→
Δ
X
is
generated
by
inertia
groups
(i.e.,
images
of
the
various
Δ
(K
X
)
x
→
Δ
U
),
and
the
action
of
Γ
K
on
an
inertia
group
is
via
the
cyclotomic
character,
i.e.,
(in
the
language
of
Hodge-Tate
Galois
representations)
of
weight
one.
It
thus
follows
(from
Lemma
2.2
(ii),
for
n
=
0,
j
=
−1)
that
the
morphism
Δ
→
Z
(
K)
U
Y
(obtained
from
the
diagram
above)
factors
through
Δ
X
.
Hence,
we
obtain
a
morphism
But
now,
it
follows
immediately
(by
the
universal
property
of
the
truncated
Δ
→
Z
(
K).
X
Y
from
Δ
Malčev
completion,
plus
“weight
arguments”)
from
the
construction
of
Z
X
(
K)
X
factors
naturally
through
Z
(
K).
that
this
morphism
Δ
→
Z
(
K)
This
shows
the
X
Y
X
existence
of
a
morphism
as
claimed
in
the
statement
of
the
Lemma.
The
fact
that
this
morphism
is
surjective
follows
from
the
fact
that
θ
is
open
(which
implies
that
the
induced
ab
morphism
Δ
ab
U
→
Δ
Y
on
abelianizations
is
open,
hence
surjective
after
tensoring
with
Q
p
).
defined
by
α
satisfies
the
condition
(∗)
spl
Lemma
5.3.
The
action
of
Γ
K
on
Z
Y
(
K)
Y
discussed
in
Section
4.
defined
by
α
satisfies
(∗)
spl
follows
Proof.
Indeed,
that
the
action
of
Γ
K
on
Z
X
(
K)
X
from
Proposition
3.5.
Thus,
Lemma
5.3
follows
from
the
surjectivity
of
the
morphism
of
Lemma
5.2.
Proposition
5.4.
Let
θ
:
Π
U
K
→
Π
Y
K
be
a
continuous
open
homomorphism
over
Γ
K
,
α
U
:
Γ
K
→
Π
U
K
a
geometric
(Definition
5.1)
section,
and
α
Y
:
Γ
K
→
Π
Y
K
the
composite
of
α
U
with
θ.
Then
α
Y
is
J
-geometric
(Definition
4.3).
Proof.
This
follows
by
combining
Lemma
5.3
with
Proposition
4.4.
Section
6:
F-Geometricity
and
FI-Geometricity
def
Let
K
be
a
finite
extension
of
Q
p
.
Let
S
=
Spec(R),
where
R
is
an
O
K
-algebra
noncanonically
isomorphic
to
O
K
[[t]]
(and
t
is
an
indeterminate).
Let
η
S
be
the
generic
34
def
def
point
of
S
(regarded
as
a
scheme).
Let
Γ
S
=
π
1
(S
K
)
(where
S
K
=
S
⊗
O
K
K,
and
the
“π
1
”
is
with
respect
to
some
base-point
which
we
omit
to
simplify
notation);
let
Γ
η
S
be
the
absolute
Galois
group
of
K(η
S
)
(the
function
field
of
S).
Thus,
Γ
S
can
be
naturally
regarded
as
a
quotient
of
Γ
η
S
.
If
H
⊆
Γ
η
S
is
an
open
subgroup
corresponding
to
some
finite
étale
covering
η
S
→
η
S
,
then
we
shall
write
Γ
η
S
for
H;
S
=
Spec(R
)
for
the
normalization
of
S
in
η
S
;
and
Γ
S
for
the
quotient
of
Γ
η
S
corresponding
to
étale
coverings
def
of
S
K
=
S
⊗
O
K
K.
Thus,
S
is
finite
and
flat
(since
S
is
regular
of
dimension
2,
and
R
is
normal,
hence
an
R-module
of
depth
2)
over
S.
Remark.
For
the
reader
familiar
with
[Mzk2],
we
remark
that
this
Section
and
the
next
are,
in
some
sense,
a
generalization
of
Section
5
of
[Mzk2],
and
play
a
comparable
role
in
the
present
paper
to
that
of
Section
5
in
[Mzk2].
For
the
reader
not
familiar
with
[Mzk2],
we
remark
that,
nevertheless,
we
do
not
assume
any
knowledge
of
[Mzk2]
in
the
following
discussion.
We
begin
by
considering
a
continuous
Z
p
[Γ
η
S
]-module
V
,
where
V
,
as
a
Z
p
-module,
is
a
finite
and
free.
Definition
6.1.
We
shall
refer
to
V
as
potentially
geometric
if
there
exists
some
open
subgroup
Γ
η
S
⊆
Γ
η
S
such
that
the
Γ
η
S
-module
obtained
by
restricting
the
Γ
η
S
-action
on
V
to
Γ
η
S
arises
as
the
Tate
module
of
some
p-divisible
group
G
→
Spec(O
K
),
where
K
⊆
R
K
is
a
finite
extension
of
K.
Note,
in
particular,
that
the
Γ
η
S
-action
on
V
then
factors
through
the
quotient
Γ
η
S
→
Γ
S
.
Moreover,
if
V
is
potentially
geometric,
then
we
may
make
the
following
construction:
Let
us
write
G
for
the
(not
necessarily
connected)
formal
group
(over
O
K
)
associated
to
the
p-divisible
group
G
→
Spec(O
K
)
of
Definition
6.1.
Thus,
we
have
a
natural
isomorphism
(of
finite
flat
group
schemes
over
O
K
)
between
the
kernels
G[p
n
]
∼
=
G[p
n
]
of
multiplication
by
p
n
(for
all
n
≥
0)
on
G
and
G.
In
particular,
if
we
consider
the
exact
sequence
(generalizing
the
Kummer
sequence,
which
corresponds
to
the
case
where
G
is
the
formal
group
associated
to
the
multiplicative
group
G
m
)
0
−→
G[p
n
]
−→
G
p
n
−→
G
−→
0
as
an
exact
sequence
of
sheaves
on
the
finite
flat
site
of
S
,
then
we
get
a
natural
map
1
(S
,
G[p
n
])
G(S
)
→
H
flat
35
(where
the
cohomology
group
is
relative
to
the
finite
flat
topology
on
S
)
for
n
≥
0.
Since
étale
morphisms
are
quasi-finite
and
flat,
we
also
have
a
natural
morphism
1
1
H
flat
(S
,
G[p
n
])
→
H
et
(S
,
G[p
n
]
K
)
Thus,
if
we
compose
the
above
two
morphisms,
take
the
inverse
limit
with
respect
to
n,
1
and
use
the
fact
that
H
et
(S
,
−)
∼
=
H
1
(Γ
S
,
−),
we
get
a
natural
morphism
κ
G
:
G(S
)
→
H
1
(Γ
S
,
V
)
which
one
may
regard
as
a
generalization
of
the
Kummer
map
(from
units
of
a
field
to
a
certain
Galois
cohomology
group
of
the
field).
Remark.
Note
that
when
the
formal
group
G
arises
from
an
abelian
variety
over
O
K
,
then
the
cohomology
class
that
one
obtains
by
applying
κ
G
to
a
point
of
G
coincides
with
the
cohomology
class
that
one
obtains
(cf.
the
discussion
of
Section
4)
by
looking
at
the
morphism
induced
on
arithmetic
fundamental
groups
by
the
corresponding
point
of
the
abelian
variety.
Indeed,
this
is
a
matter
of
general
nonsense
–
cf.,
e.g.,
[Naka2],
Claim
(2.2);
[NTs],
Lemma
(4.14).
Definition
6.2.
Suppose
that
V
is
potentially
geometric,
and
that
Γ
η
S
⊆
Γ
η
S
is
as
in
Definition
6.1.
Then
we
define
def
H
f
1
(S
K
,
V
)
⊆
H
1
(S
K
,
V
)
=
H
1
(Γ
S
,
V
)
to
be
the
subset
of
elements
ζ
∈
H
1
(S
K
,
V
)
such
that
some
nonzero
multiple
of
ζ
lies
in
the
image
of
G(S
)
under
the
morphism
κ
G
.
Now
let
X
η
S
→
η
S
be
a
proper
hyperbolic
curve
over
η
S
such
that
for
some
open
Γ
η
S
⊆
Γ
η
S
,
and
some
proper,
hyperbolic
curve
Z
→
Spec(K
)
(where
K
⊆
R
K
is
a
finite
extension
of
K,
and
we
assume
that
Z(K
)
=
∅),
we
have
an
isomorphism
of
η
S
-curves
X
η
S
×
η
S
η
S
∼
=
Z
×
K
η
S
Then
we
make
the
following
technical
Definition
6.3.
We
shall
call
X
η
S
irreducibly
splittable
if
for
some
open
Γ
η
S
⊆
Γ
η
S
,
and
some
proper,
hyperbolic
curve
Z
→
Spec(K
)
(where
K
⊆
R
K
is
a
finite
extension
of
K,
and
we
assume
that
Z(K
)
=
∅),
we
have
an
isomorphism
of
η
S
-curves
X
η
S
×
η
S
η
S
∼
=
Z
×
K
η
S
36
and,
moreover,
the
(induced)
map
S
→
Spec(O
K
)
satisfies:
(i)
S
→
Spec(O
K
)
admits
a
section;
(ii)
the
morphism
S
→
Spec(O
K
)
has
geometrically
irreducible
fibers.
As
usual,
we
have
an
exact
sequence
1
→
Δ
X
η
=
H
X
→
Π
X
η
→
Γ
η
S
→
1
S
S
where
we
write
H
X
for
the
abelianization
of
Δ
X
η
.
Thus,
H
X
has
a
natural
structure
of
S
continuous
Γ
η
S
-module.
Moreover,
as
a
Z
p
-module,
it
is
free
of
rank
2g
X
(where
g
X
is
the
genus
of
X
η
S
).
Let
us
write
I
H
X
→
H
X
for
the
quotient
of
H
X
by
all
elements
of
H
X
on
which
some
open
subgroup
of
Γ
η
S
acts
via
I
the
cyclotomic
character.
(The
“I”
comes
from
the
fact
that
H
X
is
obtained
by
forming
the
quotient
of
H
X
by
all
of
its
“inertia-like”
subgroups.)
Next,
let
F
⊆
H
X
H
X
F
be
the
largest
Γ
η
S
-submodule
of
H
X
which
has
no
nonzero
torsion-free
quotients
H
X
→
Q
such
that
some
open
subgroup
of
Γ
η
S
acts
trivially
on
Q.
(Here,
the
“F
”
stands
for
F
“finite.”
This
is
because
H
X
corresponds
to
the
portion
of
H
X
that
(potentially)
extends
to
a
p-divisible
group
–
i.e.,
a
direct
limit
of
finite
flat
group
schemes
–
over
O
K
.)
Then
F
I
it
is
well-known
(see,
e.g.,
[FC],
Chapter
III)
that
H
X
and
H
X
are
Cartier-dual
to
one
F
another,
and,
moreover,
that
H
X
is
potentially
geometric
(Definition
6.1).
(Indeed,
in
[FC],
Chapter
III,
one
finds
a
discussion
of
how
one
may
obtain
(in
a
natural
fashion)
abelian
varieties
over
K
as
“quotients”
(by
some
group
of
periods)
of
semi-abelian
varieties
that
(potentially)
extend
over
O
K
in
such
a
way
that
the
dimensions
of
their
toral
parts
are
the
same
over
the
generic
and
special
points
of
O
K
.
If
we
apply
this
theory
to
the
Jacobian
J
Z
F
of
the
curve
Z,
then
H
X
is
the
Tate
module
of
this
semi-abelian
variety
(that
potentially
I
F
I
is
the
Cartier
dual
of
H
X
,
it
thus
follows
that
H
X
is
also
extends
over
O
K
).)
Since
H
X
potentially
geometric.
Let
FI
I
H
X
⊆
H
X
F
I
F
FI
FI
be
the
image
of
H
X
in
H
X
.
Thus,
we
have
a
surjection
H
X
→
H
X
,
and
H
X
is
also
potentially
geometric.
Finally,
let
def
M
I
F
H
X
=
Ker(H
X
→
H
X
)
⊆
H
X
37
M
(Here,
the
“M”
stands
for
“multiplicative.”
This
is
because
H
X
arises
from
the
portion
F
that
corresponds
to
twisted
copies
of
(the
Tate
module
of)
the
multiplicative
group
of
H
X
M
G
m
.)
H
X
is
also
potentially
geometric.
Next,
we
would
like
to
consider
a
section
α
:
Γ
η
S
→
Π
X
η
of
Π
X
η
→
Γ
η
S
.
Now
recall
S
S
the
isomorphism
X
η
S
×
η
S
η
S
∼
=
Z
×
K
η
S
By
composing
α|
Γ
η
:
Γ
η
S
→
Π
X
η
with
the
projection
Π
X
η
→
Π
Z
induced
by
this
S
S
S
isomorphism,
we
thus
obtain
a
morphism
Γ
η
S
→
Π
Z
,
whose
composite
with
Π
Z
→
Γ
K
is
the
natural
morphism
Γ
η
S
→
Γ
K
.
Now
let
us
make
the
following
assumption
on
α:
(∗)
S
This
morphism
Γ
η
S
→
Π
Z
factors
through
Γ
S
.
Let
us
denote
the
resulting
morphism
by
β
:
Γ
S
→
Π
Z
.
Now
let
γ
:
Γ
S
→
Π
Z
be
any
morphism
obtained
by
composing
the
natural
morphism
Γ
S
→
Γ
K
with
some
section
Γ
K
→
Π
Z
arising
from
a
geometric
point
∈
Z(K
).
Let
β
J
,
γ
J
:
Γ
S
→
Π
(1)
be
the
morphisms
obtained
by
composing
β
and
γ,
respectively,
with
J
Z
Π
Z
→
Π
(1)
.
Then
the
difference
β
J
−
γ
J
defines
an
element
J
Z
,
H
Z
)
δ
Z
∈
H
1
(S
K
hence
an
element
,
H
X
)
δ
X
∈
H
1
(S
K
I
I
whose
image
in
H
1
(S
K
,
H
X
)
we
denote
by
δ
X
.
Definition
6.4.
Suppose
that
α
:
Γ
η
S
→
Π
X
η
is
a
section
that
satisfies
the
assump-
S
tion
(∗)
S
above.
Then
we
shall
call
α
F
-geometric
(respectively,
F
I-geometric)
if
some
I
F
nonzero
multiple
of
δ
X
(respectively,
δ
X
)
lies
in
the
image
of
H
f
1
(S
K
,
H
X
)
(respectively,
1
FI
1
1
I
H
f
(S
K
,
H
X
))
in
H
(S
K
,
H
X
)
(respectively,
H
(S
K
,
H
X
)).
Note
that
if
α
is
F
-geometric,
it
is
also
automatically
F
I-geometric.
Also,
let
us
observe
that
the
definition
of
“F
-geometric”
or
“F
I-geometric”
is
independent
of
the
choice
of
γ
(cf.
Lemma
4.1
and
the
discussion
preceding
it).
Indeed,
to
see
this,
it
suffices
to
verify
that
some
nonzero
multiple
of
the
class
∈
H
1
(S
K
,
H
X
)
arising
from
the
difference
of
two
1
F
γ’s
lies
in
H
f
(S
K
,
H
X
).
But
the
difference
of
two
γ’s
(both
of
which
arise
from
a
geometric
point
∈
Z(K
))
defines
a
point
∈
J
Z
(K
)
(where
J
Z
is
the
Jacobian
of
Z).
Moreover,
since
the
residue
field
of
K
is
finite,
it
follows
that
some
nonzero
multiple
of
this
point
in
J
Z
(K
)
38
extends
to
an
O
K
-valued
point
of
the
Néron
model
J
Z
of
J
Z
over
O
K
which
maps
the
special
point
of
Spec(O
K
)
to
the
identity
of
J
Z
.
On
the
other
hand,
since
G
F
–
i.e.,
the
F
“G”
(cf.
the
discussion
following
Definition
6.1)
for
the
potentially
geometric
module
H
X
–
is
simply
the
formal
group
defined
by
J
Z
,
it
thus
follows
that
such
a
point
∈
J
Z
(O
K
)
defines
a
point
in
G
F
(O
K
)
⊆
G
F
(S
).
Thus,
by
Definition
6.2,
we
see
that
the
difference
F
between
the
two
γ’s
lies
in
H
f
1
(S
K
,
H
X
),
as
desired.
Proposition
6.5.
Suppose
that
X
S
is
irreducibly
splittable
(cf.
Definition
6.3).
Then
any
F
I-geometric
α
is
also
F
-geometric.
Proof.
First,
let
us
state
that
throughout
the
proof,
“S
”
will
be
a
fixed
S
satisfying
the
conditions
of
Definition
6.3.
Next,
let
us
observe
that
the
cokernel
of
the
natural
morphism
F
FI
H
f
1
(S
K
,
H
X
)
→
H
f
1
(S
K
,
H
X
)
is
torsion.
Indeed,
this
follows
from
Definition
6.2
and
the
fact
that
if
G
F
and
G
F
I
are
the
respective
formal
groups
as
in
Definition
6.2,
then
the
natural
morphism
G
F
→
G
F
I
is
formally
smooth
(hence
surjective
on
S
-valued
points).
I
,
H
X
)
whose
image
ζ
I
∈
H
1
(S
K
,
H
X
)
lies
Thus,
if
we
start
with
a
class
ζ
∈
H
1
(S
K
1
FI
in
the
image
of
H
f
(S
K
,
H
X
),
then
(after
replacing
ζ
by
a
nonzero
multiple
of
ζ),
we
,
H
X
)
such
that:
(a.)
ζ
lies
in
the
image
may
assume
that
there
exists
a
ζ
∈
H
1
(S
K
1
F
1
I
of
H
f
(S
K
,
H
X
);
(b.)
ζ
−
ζ
maps
to
0
in
H
(S
K
,
H
X
).
Moreover,
by
the
definition
of
def
M
I
H
X
=
Ker(H
X
→
H
X
),
it
follows
that
(b.)
may
be
rewritten
in
the
form:
“ζ
−
ζ
lies
in
M
the
image
of
H
1
(S
K
,
H
X
)
in
H
1
(S
K
,
H
X
).”
With
these
observations
in
hand,
it
follows
that
it
suffices
to
prove
that
M
(∗)
im
The
image
of
H
1
(S
K
,
H
X
)
in
H
1
(S
K
,
H
X
)
is
contained
up
to
torsion
(i.e.,
up
to
multiplication
by
a
nonzero
integer)
in
the
image
of
F
,
H
X
)
in
H
1
(S
K
,
H
X
).
H
f
1
(S
K
Note
first
that
by
replacing
K
by
a
finite
extension
of
K
(and
thus
also
enlarging
S
–
note
that
this
does
not
affect
the
validity
of
the
conditions
(i)
and
(ii)
of
Definition
6.3),
M
we
may
assume
that
the
action
of
Γ
S
on
H
X
(−1)
(where
the
“(−1)”
is
a
Tate
twist)
is
M
trivial.
(Thus,
in
particular,
as
a
Γ
S
-module,
H
X
is
isomorphic
to
a
direct
sum
of
a
finite
number
of
copies
of
Z
p
(1).)
On
the
other
hand,
by
Lemma
6.6
below
(and
the
Kummer
,
Z
p
(1))
=
{(R
K
)
×
}
∧
(where
the
“∧”
denotes
p-adic
exact
sequence),
it
follows
that
H
1
(S
K
,
Z
p
(1))
=
{(R
)
×
}
∧
and
(the
image
of)
completion)
is
generated
up
to
torsion
by
H
f
1
(S
K
M
,
H
X
)
is
generated
up
to
torsion
H
1
(K
,
Z
p
(1))
=
{(K
)
×
}
∧
.
Thus,
it
follows
that
H
1
(S
K
1
M
1
M
by
H
f
(S
K
,
H
X
)
and
(the
image
of)
H
(K
,
H
Z
).
Next,
observe
that
the
image
of
H
1
(K
,
H
Z
M
)
in
H
1
(K
,
H
Z
)
is
contained
up
to
torsion
in
the
image
of
H
f
1
(K
,
H
Z
F
)
in
H
1
(K
,
H
Z
).
Indeed,
this
follows
from
the
theory
of
[BK],
39
§3
(cf.
the
proof
of
Lemma
4.1).
Namely,
since
H
Z
M
is
“of
weight
one,”
the
image
of
is
zero,
so
any
cohomology
classes
H
M
⊆
H
in
the
weight
zero
component
of
H
⊗
K
Z
Z
1
Z
But
M
1
(K
,
H
Z
)
go
to
zero
in
H
(K
,
−)
of
the
weight
zero
component
of
H
Z
⊗
Z
p
K.
in
H
this
means
(as
we
saw
in
Lemma
4.1),
that
such
cohomology
classes
arise
geometrically,
as
desired.
(Note
that,
by
the
above
argument
involving
Néron
models
(in
the
discussion
immediately
preceding
Proposition
6.5),
it
follows
that
(relative
to
pulling
back
cohomology
classes
over
K
to
cohomology
classes
over
S
K
),
the
notation
“H
f
1
(S
K
,
−)”
of
Definition
1
6.2
is
consistent
with
the
notation
“H
f
(K
,
−)”
of
[BK],
§3.)
Putting
everything
together,
we
thus
see
that
(∗)
im
has
been
verified.
This
completes
the
proof
of
the
Proposition.
Lemma
6.6.
For
S
satisfying
condition
(ii)
of
Definition
6.3,
we
have
that
(R
K
)
×
is
generated
up
to
torsion
by
(R
)
×
and
(K
)
×
.
Proof.
Indeed,
condition
(ii)
of
Definition
6.3
implies
that
if
π
is
a
uniformizing
element
of
O
K
⊆
R
,
then
the
ideal
π
·
R
is
contained
in
a
unique
prime
ideal
of
height
one
of
R
.
Let
us
denote
this
prime
ideal
by
℘.
Let
x
∈
R
be
an
element
which
becomes
a
unit
in
R
K
.
Then
x
is
invertible
at
every
height
one
prime
of
R
except
(possibly)
℘.
Moreover,
by
replacing
x
by
some
x
n
(where
n
is
independent
of
x),
we
may
assume
that
x
has
the
same
valuation
as
π
m
(for
some
nonnegative
integer
m)
in
the
discrete
valuation
ring
R
℘
.
It
thus
follows
that
π
−m
·
x
is
a
unit
at
every
height
one
prime
of
R
.
Since
R
is
normal,
this
implies
that
π
−m
·
x
∈
(R
)
×
.
This
completes
the
proof
of
the
Lemma.
Section
7:
From
F-Geometricity
to
Line
Bundles
In
this
Section,
we
use
some
elementary
algebraic
geometry
(Lemmas
7.1
and
7.2)
to
translate
the
rather
abstract
and
technical
condition
of
“F
-geometricity”
into
a
more
tractable
existence
criterion
(Proposition
7.4)
for
line
bundles.
Let
S
be
the
spectrum
of
a
field
of
characteristic
zero.
Let
X
→
S
be
a
proper
hyperbolic
curve
of
genus
g
over
S.
Let
(N
)
N
be
a
fixed
positive
integer.
We
would
like
to
consider
the
natural
morphism
X
→
J
X
(given
by
mapping
a
point
x
of
X
to
the
line
bundle
O
X
(N
·
x)).
Taking
the
product
of
(N
)
this
morphism
with
X
(on
the
right),
we
obtain
a
morphism
ξ
:
X
×
S
X
→
J
X
×
S
X.
Lemma
7.1.
(N
)
There
exists
a
line
bundle
on
J
X
×
S
X
whose
pull-back
via
ξ
is
a
nonzero
def
tensor
power
of
the
line
bundle
D
=
O
X×
S
X
(Δ)
(where
Δ
⊆
X
×
S
X)
on
X
×
S
X.
Proof.
First,
let
us
take
the
product
of
ξ
with
one
more
copy
of
X
on
the
right,
to
(N
)
obtain
a
morphism
ξ
:
X
×
S
X
×
S
X
→
J
X
×
S
X
×
S
X.
For
i,
j
=
1,
2,
3
such
that
40
i
=
j,
let
Δ
ij
⊆
X
×
S
X
×
S
X
denote
the
diagonal
given
by
setting
equal
the
i
th
and
j
th
components
of
the
triple
product.
Let
D
ij
denote
the
line
bundle
O
X×
S
X×
S
X
(Δ
ij
).
In
the
triple
product
of
X,
we
consider
the
second
X
to
be
the
“true
curve
X”;
the
first
X
to
be
a
“parameter
space
for
the
family
of
line
bundles
D
12
”
(regarded
as
line
bundles
on
the
“true
curve”
given
by
the
second
factor);
and
the
third
X
to
be
a
“base
extension
X
→
S.”
Over
this
extended
base,
the
“true
curve”
acquires
a
“section”
Δ
23
.
Thus,
it
follows
from
the
general
theory
of
the
Picard
functor
of
a
family
of
curves
for
which
a
(N
)
section
exists
that
there
exists
a
line
bundle
L
on
J
X
×
S
X
×
S
X
whose
pull-back
via
ξ
−1
⊗N
is
equal
to
E
=
(D
12
⊗
D
13
)
.
def
Now
let
us
consider
the
“determinant
of
the
higher
direct
image
sheaves”
(cf.
[MB],
§1,
for
an
exposition
of
this
notion)
of
L
and
E
for
the
morphisms
X
×
S
X
×
S
X
→
X
×
S
X
(N
)
(N
)
and
J
X
×
S
X
×
S
X
→
J
X
×
S
X
given
by
forgetting
the
third
factor.
We
denote
the
respective
“determinants
of
the
higher
direct
image
sheaves”
of
L
and
E
by
L
and
E.
Thus,
(by
the
functoriality
of
forming
the
“determinant
of
the
higher
direct
image
sheaves”)
L
(N
)
is
a
line
bundle
on
J
X
×
S
X
such
that
ξ
∗
(L)
=
E.
Moreover,
I
claim
that
we
can
write
E
=
D
⊗N
⊗
F
⊗m
(for
m
∈
Z)
where
F
is
a
line
bundle
on
X
×
S
X
obtained
by
pulling
back
some
ω
X/S
from
the
first
factor
of
X
×
S
X,
and
tensoring
with
the
pull-back
to
X
×
S
X
of
some
line
bundle
M
on
S.
Indeed,
the
“D
⊗N
-term”
arises
from
the
fact
that
D
12
is
the
pull-
back
of
D
via
the
projection
X
×
S
X
×
S
X
→
X
×
S
X
under
consideration.
Then
the
difference
between
the
“determinants
of
the
higher
direct
image
sheaves”
of
the
line
−N
bundles
D
13
=
O
X×
S
X×
S
X
(−N
·Δ
13
)
and
O
X×
S
×
S
X
may
be
computed
using
the
natural
inclusion
−N
=
O
X×
S
X×
S
X
(−N
·
Δ
13
)
⊆
O
X×
S
×
S
X
D
13
By
using
the
fact
that
the
“determinant
of
the
higher
direct
image
sheaves”
is
multiplicative
on
exact
sequences,
we
thus
obtain
that
this
difference
is
equal
to
some
power
of
the
pull-
back
to
X
×
S
X
(via
the
projection
to
the
first
factor)
of
O
X×
S
X
(−Δ)|
Δ=X
=
ω
X/S
.
On
the
other
hand,
the
“determinant
of
the
higher
direct
image
sheaves”
of
the
trivial
line
bundle
on
X
×
S
X
×
S
X
is
clearly
the
pull-back
(to
X
×
S
X)
of
a
line
bundle
M
on
S.
This
gives
us
a
line
bundle
F
of
the
form
discussed
above,
hence
completes
the
proof
of
the
claim.
By
replacing
L
with
L
⊗
O
S
M
−1
,
we
may
assume
that
M
is
trivial.
Moreover,
by
(N
)
Lemma
7.2
below,
there
exists
a
line
bundle
P
on
J
X
whose
pull-back
to
X
(via
the
(N
)
natural
morphism
X
→
J
X
)
is
some
nonzero
tensor
power
of
ω
X/S
.
Thus,
replacing
L
by
a
tensor
product
of
appropriate
powers
of
L
and
(P|
J
(N
)
×
S
X
)
completes
the
proof
of
X
the
Lemma.
41
(N
)
Lemma
7.2.
There
exists
a
line
bundle
P
on
J
X
tensor
power
of
ω
X/S
.
whose
pull-back
to
X
is
a
nonzero
Proof.
Let
us
first
consider
the
case
N
=
g
−
1.
In
this
case,
we
take
P
to
be
the
line
(g−1)
(g−1)
bundle
on
J
X
defined
by
the
natural
theta
divisor
on
J
X
(i.e.,
the
image
of
the
(g−1)
natural
morphism
from
the
(g
−
1)-fold
product
of
X
to
J
X
).
Then
the
fact
that
the
⊗
1
g(g−1)
2
pull-back
of
P
to
X
is
equal
to
ω
X/S
is
an
immediate
consequence
of
[MB],
Corollary
2.5:
in
the
notation
of
loc.
cit.,
we
are
interested
here
in
the
case
where
one
takes
n
=
0;
S
=
X;
and
a
:
S
→
X
to
be
the
identity;
then
specializing
the
formula
of
loc.
cit.
to
(a)
the
zero
section
of
the
Jacobian
(note
that
the
restriction
of
“U
n
”
in
loc.
cit.
to
the
zero
section
of
the
Jacobian
is
trivial)
proves
the
assertion
concerning
the
pull-back
of
P
to
X.
Next,
let
us
consider
the
case
where
N
is
divisible
by
g
−
1.
Then
the
morphism
(N
)
(g−1)
(N
)
X
→
J
X
factors
through
J
X
→
J
X
.
Moreover,
it
follows
from
the
basic
theory
of
line
bundles
on
abelian
varieties
(see,
e.g.,
[AV],
§23,
the
Corollary
to
Theorem
2
on
p.
231)
that
a
nonzero
tensor
power
of
the
“P”
considered
in
the
previous
paragraph
descends
(g−1)
(N
)
from
J
X
to
J
X
.
This
completes
the
proof
in
the
case
where
N
is
divisible
by
g
−
1.
Finally,
we
consider
the
case
of
arbitrary
positive
N
.
In
this
case,
we
have
a
natural
(N
)
(N
(g−1))
map
J
X
→
J
X
(multiplication
by
g
−
1).
But,
by
the
preceding
paragraph,
we
(N
(g−1))
(N
)
.
Thus,
by
pulling
this
line
bundle
back
to
J
X
,
already
have
a
suitable
“P”
on
J
X
(N
)
we
obtain
a
suitable
“P”
on
J
X
.
This
completes
the
proof
of
the
Lemma.
def
Next,
we
would
like
to
consider
a
section
α
:
Γ
S
=
π
1
(S)
→
Π
X
of
Π
X
→
Γ
S
.
By
(N
)
composing
α
with
π
1
(−)
applied
to
X
→
J
X
,
we
obtain
a
section
α
N
J
:
Γ
S
→
Π
(N
)
.
J
X
(N
)
Lemma
7.3.
Suppose
that
α
N
J
arises
from
a
geometric
section
∈
J
X
(S).
Then
there
exists
a
line
bundle
on
X
of
degree
prime
to
p.
Proof.
By
taking
the
fibered
product
(over
Γ
S
)
of
α
with
the
identity
on
Π
X
,
we
obtain
a
morphism
α
X
:
Π
X
→
Π
X×
S
X
.
Since
S
is
the
spectrum
of
the
field,
the
group
cohomology
of
Π
X×
S
X
computes
the
p-adic
étale
cohomology
of
X
×
S
X
(cf.
Lemma
0.4).
Thus,
we
can
form
the
arithmetic
first
Chern
class
of
the
line
bundle
D
of
Lemma
7.1
(cf.
def
Definition
0.3):
c
1
(D)
∈
H
2
(Π
X×
S
X
,
Z
p
(1)).
Let
ζ
=
α
∗
X
(c
1
(D))
∈
H
2
(Π
X
,
Z
p
(1)).
On
the
other
hand,
by
composing
α
X
with
π
1
(−)
of
the
morphism
ξ,
we
obtain
a
morphism
.
Moreover,
by
assumption,
α
N
α
N
ξ
:
Π
X
→
Π
(N
)
ξ
arises
from
a
geometric
morphism
J
X
×
S
X
(N
)
(N
)
∗
X
→
J
X
×
S
X.
Thus,
if
L
is
a
line
bundle
on
J
X
×
S
X,
we
obtain
that
(α
N
ξ
)
(c
1
(L))
∈
H
2
(Π
X
,
Z
p
(1))
can
be
written
as
c
1
(M),
for
some
line
bundle
M
on
X.
Now
recall
that
(N
)
by
Lemma
7.1,
there
exists
a
line
bundle
L
on
J
X
×
S
X
such
that
ξ
∗
L
is
a
nonzero
tensor
power
of
D.
Thus,
putting
everything
together,
it
follows
that
some
nonzero
multiple
of
42
the
abstract
cohomology
class
ζ
∈
H
2
(Π
X
,
Z
p
(1))
can
be
written
in
the
form
c
1
(M)
for
some
line
bundle
M
on
X,
i.e.,
m
·
ζ
=
c
1
(M)
(for
some
nonzero
m
∈
Z).
Next,
let
us
recall
that
since
ζ
was
constructed
from
D,
the
image
of
ζ
under
the
“degree
map”
H
2
(Π
X
,
Z
p
(1))
→
H
2
(Δ
X
,
Z
p
(1))
∼
=
Z
p
is
equal
to
1,
so
deg(M)
=
m.
Thus,
we
can
argue
as
follows
(cf.
[Mzk2],
Lemma
6.1):
Write
m
=
a
·
p
b
,
where
a
is
an
integer
prime
to
p,
and
b
is
a
nonnegative
integer.
Thus,
c
1
(M)
vanishes
in
H
2
(Π
X
,
(Z/p
b
Z)(1)).
But,
by
the
Kummer
exact
sequence,
this
implies
that
there
exists
a
line
bundle
P
on
X
such
that
b
P
⊗p
∼
=
M.
In
particular,
deg(P)
=
p
−b
·
deg(M)
=
a,
so
the
existence
of
P
completes
the
proof
of
the
Lemma.
Now
let
us
consider
the
following
situation:
Let
K
be
a
finite
extension
of
Q
p
.
Let
X
K
→
Spec(K)
and
Y
K
→
Spec(K)
be
proper
hyperbolic
curves
over
K.
Let
U
K
be
the
generic
point
of
X
K
.
Moreover,
let
us
assume
that
we
have
been
given
a
continuous
open
homomorphism
over
Γ
K
θ
:
Π
U
K
→
Π
Y
K
Now
let
S,
η
S
be
as
at
the
beginning
of
Section
6.
Write
X
η
S
for
X
K
×
K
η
S
(and
similarly,
for
U
η
S
,
Y
η
S
).
Then
by
base-change
(note
that
Π
U
η
=
Π
U
K
×
Γ
K
Γ
η
S
;
Π
Y
η
=
Π
Y
K
×
Γ
K
S
S
Γ
η
S
),
θ
induces
a
continuous
open
homomorphism
θ
η
S
:
Π
U
η
→
Π
Y
η
S
S
Let
Y
η
S
→
Y
η
S
be
a
finite
étale
covering
induced
by
some
open
subgroup
of
Π
Y
η
that
S
surjects
onto
Γ
η
S
.
Note
that
θ
η
S
allows
us
to
define
the
pull-back
of
Y
η
S
→
Y
η
S
to
U
η
S
.
Let
U
η
S
→
U
η
S
be
a
connected
component
of
this
pull-back
which
is
geometrically
connected
over
η
S
.
Note
that
U
η
S
→
U
η
S
extends
to
a
finite
(possibly
ramified)
covering
X
η
S
→
X
η
S
.
Thus,
X
η
S
is
a
proper
hyperbolic
curve
over
η
S
.
Let
us
suppose
that
Y
η
S
is
irreducibly
splittable
(Definition
6.3).
Then
it
follows
immediately
that
X
η
S
is
also
irreducibly
splittable.
(Indeed,
the
definition
of
“irreducibly
splittable”
only
involves
the
base
(i.e.,
S,
S
,
etc.),
plus
finite
étale
coverings
of
the
curve,
i.e.,
it
may
be
phrased
entirely
in
terms
of
(the
base
plus)
fundamental
groups.
Thus,
the
fact
that
X
η
S
and
Y
η
S
are
related
by
θ
η
S
is
enough
to
guarantee
that
“Y
η
S
irreducibly
splittable
=⇒
X
η
S
irreducibly
splittable.”)
Next,
we
would
like
to
consider
a
section
α
U
:
Γ
η
S
→
Π
U
η
S
of
Π
U
η
→
Γ
η
S
.
Composing
α
U
with
π
1
(−)
of
the
natural
morphism
U
η
S
→
X
η
S
gives
rise
S
to
a
section
α
X
:
Γ
η
S
→
Π
X
η
.
Composing
α
U
with
the
morphism
θ
η
S
:
Π
U
η
→
Π
Y
η
S
S
S
induced
by
θ
η
S
gives
a
section
α
Y
:
Γ
η
S
→
Π
Y
η
.
S
43
The
following
key
result
is
the
culmination
of
our
efforts
in
Sections
1
through
7:
Proposition
7.4.
Under
the
circumstances
just
described
above,
let
us
assume
that
α
X
is
F
I-geometric
(Definition
6.4).
Then
it
follows
that
Y
η
S
admits
a
line
bundle
of
degree
prime
to
p.
Proof.
First
observe
that
the
morphism
Δ
U
η
→
Δ
Y
η
→
H
Y
I
η
(where
H
Y
I
η
is
as
in
the
S
S
S
S
I
discussion
preceding
Definition
6.4)
factors
through
H
X
.
(Indeed,
this
follows
by
observ-
η
S
ing,
as
in
the
proof
of
Lemma
5.2,
that
the
inertia
groups
of
Δ
U
η
map
to
zero
in
H
Y
I
η
.)
S
S
I
Thus,
we
obtain
a
Γ
η
S
-equivariant
morphism
H
X
→
H
Y
I
η
.
In
particular,
it
follows
by
ηS
S
using
Tate’s
theorem
(i.e.,
Theorem
4
of
[Tate]
–
that
morphisms
between
Tate
modules
of
p-divisible
groups
induce
morphisms
between
the
p-divisible
groups,
hence
morphisms
FI
1
FI
between
the
respective
formal
groups)
that
H
f
1
(S
K
,
H
X
)
maps
to
H
f
(S
K
,
H
Y
),
for
any
ηS
ηS
finite
étale
covering
η
S
→
η
S
of
η
S
as
in
Definition
6.2.
Next,
I
claim
that
α
Y
is
F
I-geometric.
Indeed,
this
follows
from
Proposition
5.4,
plus
the
observation
of
the
preceding
paragraph.
In
words:
This
claim
amounts
to
the
claim
(cf.
Definition
6.4)
that
the
“J
-portion”
of
α
Y
(i.e.,
the
result
of
composing
α
Y
with
(1)
Π
Y
η
→
Π
(1)
,
where
J
Y
is
the
Picard
scheme
of
line
bundles
of
degree
1
on
Y
η
S
)
differs
J
Y
S
from
the
“J
-portion”
of
a
constant
(i.e.,
arising
from
a
point
defined
over
a
finite
extension
of
K
of
the
curve
“Z”
of
the
discussion
of
Section
6)
geometric
section
of
Π
Y
η
by
a
class
S
in
H
f
1
(S
K
,
H
Y
F
η
I
).
On
the
other
hand,
since
we
are
operating
under
the
assumption
that
S
α
X
is
F
I-geometric,
we
know
that
the
J
-portion
of
α
X
differs
from
the
J
-portion
of
a
FI
constant
geometric
section
of
Π
X
η
by
a
class
in
H
f
1
(S
K
,
H
X
).
Moreover,
by
Proposi-
η
S
S
tion
5.4,
constant
geometric
sections
of
Π
U
η
map
to
constant
J
-geometric
(which
is
as
S
good
as
“geometric”
for
us,
since
we
are
only
interested
in
“J
-portions”
here)
sections
of
FI
,
H
X
Π
Y
η
.
Thus,
since
(by
the
observation
of
the
preceding
paragraph)
H
f
1
(S
K
)
maps
η
S
S
to
H
f
1
(S
K
,
H
Y
F
η
I
),
we
conclude
that
α
Y
is
F
I-geometric,
as
desired.
This
completes
the
S
proof
of
the
claim.
Now
since
Y
η
S
is
also
assumed
to
be
irreducibly
splittable,
it
follows
from
Proposition
6.5
that
α
Y
is,
in
fact,
F
-geometric.
(N
)
Now
let
us
consider
the
morphism
α
N
J
:
Γ
η
S
→
Π
(N
)
(where
J
Y
J
Y
is
the
Picard
scheme
of
line
bundles
of
degree
N
on
Y
η
S
),
for
some
N
>
0,
induced
by
composing
α
Y
with
Π
Y
η
→
Π
(N
)
.
Note
that
the
F
-geometricity
of
α
Y
is,
by
definition,
a
property
J
Y
1
concerning
α
J
.
By
the
definition
of
F
-geometricity
(Definition
6.4),
it
follows
immediately
(N
)
that
if
N
is
large
enough,
then
α
N
J
arises
from
a
geometric
section
J
Y
(S
K
).
Indeed,
the
S
large
N
is
to
take
care
of
the
phrase
“nonzero
multiple
of”
in
Definitions
6.2
and
6.4,
plus
the
fact
that
a
priori
the
geometric
point
whose
existence
is
guaranteed
by
Definition
6.2
is
only
defined
over
η
S
,
so
we
may
need
to
apply
the
norm
map
(for
η
S
→
η
S
)
to
get
a
44
geometric
point
over
η
S
,
which
may
cause
N
to
increase
again.
Now,
applying
Lemma
7.3
completes
the
proof
of
the
Proposition.
Remark.
Note
that
Sections
6
and
7
are
made
much
more
technically
difficult
by
the
fact
that
we
start
with
a
morphism
Π
U
K
→
Π
Y
K
rather
than
a
morphism
Π
X
K
→
Π
Y
K
.
Indeed,
the
whole
business
of
distinguishing
“F
-geometric”
from
“F
I-geometric”
arises
because
one
does
not
have
a
proper
theory
of
“p-divisible
groups
of
infinite
rank,”
which
is
the
sort
of
object
that
one
must
deal
with
if
one
tries
to
work
directly
with
H
U
(the
abelianization
of
Δ
U
)
without
passing
to
H
U
I
.
Similar
technical
problems
(arising
from
the
fact
that
H
U
is
of
infinite
rank)
also
are
the
reason
behind
the
lengthiness
of
Sections
11
and
12.
Section
8:
From
Line
Bundles
to
Tame
Points
In
this
Section,
which
is
something
of
an
appendix
to
that
portion
of
the
paper
con-
stituted
by
Sections
1
through
7,
we
again
apply
some
elementary
algebraic
geometry,
this
time
to
pass
from
line
bundles
to
rational
points
of
the
curve
defined
over
tamely
ramified
extensions
of
the
given
field.
Thus,
let
M
be
a
p-adic
field
whose
residue
field
k
M
is
iso-
morphic
to
k((t)),
where
k
is
a
finite
field,
and
t
is
an
indeterminate.
Let
X
M
be
a
proper
hyperbolic
curve
over
M.
Then
we
have
the
following
result:
Proposition
8.1.
Suppose
that
X
M
admits
a
line
bundle
L
of
degree
prime
to
p.
Then
there
exists
a
tamely
ramified
extension
M
of
M
such
that
X
M
(M
)
is
nonempty.
Proof.
By
replacing
L
by
an
appropriate
(prime
to
p)
tensor
power
of
the
original
L,
we
may
assume
that
L
is
very
ample.
Thus,
by
Bertini’s
Theorem,
it
follows
that
there
exists
a
divisor
D
⊆
X
M
such
that
L
∼
=
O
X
M
(D),
and
D
is
étale
over
M.
Let
us
write
D
=
r
D
i
i=1
where
each
D
i
=
Spec(M
i
).
Thus,
M
i
is
a
finite
field
extension
of
M.
Since
deg(D)
is
prime
to
p,
it
follows
that
at
least
one
of
the
M
i
(say,
M
1
)
is
such
that
[M
1
:
M]
is
prime
to
p.
Let
M
c
be
the
Galois
closure
of
M
1
over
M.
Then
it
follows
from
the
elementary
theory
of
p-adic
fields
(in
particular,
the
fact
that
the
wild
inertia
subgroup
of
Γ
M
is
a
normal
pro-p
subgroup
of
Γ
M
,
hence
contained
in
any
Sylow-pro-p
subgroup
of
Γ
M
–
see,
e.g.,
[Ser2],
Chapter
IV,
§2)
that
M
c
is
tamely
ramified
over
M.
Thus,
in
particular,
M
1
is
tamely
ramified
over
M,
and
X
M
(M
1
)
is
nonempty.
This
completes
the
proof
of
the
Proposition.
45
Next,
before
continuing,
we
would
like
to
discuss
an
auxiliary
Lemma
that
will
be
useful
when
we
apply
Proposition
8.1.
Let
us
suppose
that
there
exists
a
p-adic
field
L
⊆
M
satisfying
the
following
properties:
(1)
We
have
an
inclusion
O
L
⊆
O
M
,
relative
to
which
m
L
·
O
M
=
m
M
.
(2)
If
k
L
is
the
residue
field
of
L,
then
k
⊆
k
L
,
and
k
L
is
a
function
field
in
one
variable
over
k.
(3)
The
inclusion
k
L
⊆
k
M
is
obtained
by
completing
the
function
field
k
L
at
one
of
its
k-valued
points.
Let
us
suppose,
moreover,
that
there
exists
a
curve
X
L
→
Spec(L)
such
that
X
M
=
X
L
⊗
L
M.
Then
we
have
the
following
result:
Lemma
8.2.
Suppose
that
there
exists
a
tamely
ramified
extension
M
of
M
of
ramifica-
tion
index
e
such
that
X
M
(M
)
is
nonempty.
Then
there
exists
a
tamely
ramified
extension
L
of
L
of
ramification
index
e
such
that
X
L
(L
)
is
nonempty.
Proof.
Since
tamely
ramified
extensions
of
M
descend
to
tamely
ramified
extensions
of
L,
it
is
easy
to
see
that
without
loss
of
generality,
we
may
assume
that
e
=
1,
and
M
=
M.
Thus,
we
must
show
that
X
L
admits
a
rational
point
over
some
unramified
extension
of
L.
Note
that
since
O
L
is
a
discrete
valuation
ring,
it
is
well-known
that
X
L
def
admits
a
regular
model
X
L
→
Spec(O
L
).
Let
X
M
=
X
L
⊗
O
L
O
M
.
Let
us
consider
the
morphism
φ
:
Spec(O
M
)
→
Spec(O
L
).
Since
finitely
generated
algebras
over
a
finite
field
are
excellent,
it
follows
that
k
M
is
separable
over
k
L
(i.e.,
the
morphism
Spec(k
M
)
→
Spec(k
L
)
is
geometrically
regular).
In
particular,
it
follows
that
φ
is
geometrically
regular.
Since
the
natural
morphism
X
M
→
X
L
is
obtained
from
φ
by
base-change,
it
thus
follows
that
it,
too,
is
geometrically
regular.
Thus,
the
regularity
of
X
L
implies
that
of
X
M
.
Now
the
fact
that
X
M
(M)
is
nonempty
(by
assumption)
implies
that
there
exist
points
in
the
special
fiber
of
X
M
at
which
X
M
is
smooth
over
O
M
.
But
this
implies
(by
descent)
that
there
exist
points
in
the
special
fiber
of
X
L
at
which
X
L
is
smooth
over
O
L
,
which,
in
turn,
implies
that
for
some
unramified
extension
L
of
L,
X
L
(L
)
is
nonempty,
as
desired.
Section
9:
Convergence
via
p-adic
Hodge
Theory
Whereas
Sections
1
through
8
formed
a
unit
devoted
essentially
to
showing
Propo-
sition
7.4,
Sections
9
and
10
form
a
new
unit,
devoted
to
showing,
by
means
of
p-adic
Hodge
theory,
that
certain
types
of
sequences
of
points
converge
p-adically
to
a
uniquely
determined
point
defined
over
a
relatively
small
field.
Thus,
in
particular,
we
shall
start
with
fresh
hypotheses,
as
follows:
46
Let
K
be
a
p-adic
local
field,
with
residue
field
k.
Let
L
be
a
p-adic
field
containing
K
whose
residue
field
k
L
is
a
function
field
in
one
variable
over
k.
Let
us
also
assume
that
k
is
algebraically
closed
in
k
L
,
and
that
m
K
·
O
L
=
m
L
.
Let
us
denote
by
def
Γ
L/K
=
Ker(Γ
L
→
Γ
K
)
=
Γ
L·K
1
the
“geometric”
fundamental
group
of
L.
Let
H
Ω
L
be
the
quotient
of
H
(Γ
L/K
,
O
L
(1))
by
its
def
torsion
submodule.
Let
X
K
→
Spec(K)
be
a
proper
hyperbolic
curve.
Let
X
L
=
X
K
⊗
K
L.
Thus,
as
usual,
we
have
a
group
extension
(obtained
by
pulling
back
1
→
Δ
X
→
Π
X
K
→
Γ
K
→
1
via
Γ
L
→
Γ
K
)
1
→
Δ
X
→
Π
X
L
→
Γ
L
→
1
Let
us
assume
that
we
are
given
a
continuous,
group
homomorphism
α
:
Γ
L
→
Π
X
K
whose
composite
with
the
projection
to
Γ
K
is
the
natural
morphism
Γ
L
→
Γ
K
.
As
usual,
α
defines
a
section
α
s
:
Γ
L
→
Π
X
L
of
Π
X
L
→
Γ
L
.
Let
us
also
assume
that
α
is
nondegenerate
in
the
sense
that
it
satisfies
the
following
group-theoretic
condition:
(∗)
non
The
natural
morphism
H
1
(Δ
X
,
Z
p
(1))
→
H
Ω
L
induced
by
α
is
nonzero.
def
Let
L
tm
be
a
maximal
tamely
ramified
extension
of
L.
Let
M
=
(L
tm
)
∧
be
its
p-adic
completion.
Since
L
tm
·
K
is
unramified
over
L
·
K,
and
Galois
cohomology
“ignores
unramified
extensions,”
we
have
that
1
(1))
=
H
1
(Γ
H
1
(Γ
L/K
,
O
L
L·K
,
O
L
(1))
=
H
(Γ
L
tm
·K
,
O
L
(1))
In
particular,
since
H
Ω
L
is
a
quotient
of
this
cohomology
module,
it
follows
from
assumption
non
(∗)
that
α|
Γ
L
tm
·K
:
Γ
L
tm
·K
→
Δ
X
is
nontrivial.
def
For
n
≥
0,
let
I
L
n
=
Im(α
s
)
·
(Δ
X
)
<n>
⊆
Π
X
L
(cf.
the
discussion
preceding
Definition
0.2
for
an
explanation
of
the
notation
“<
n
>”).
Let
ψ
n
:
X
L
n
→
X
L
be
the
corresponding
finite
étale
covering.
Let
X
L
∞
→
X
L
be
the
inverse
limit
of
the
X
L
n
.
In
this
Section,
we
would
like
to
prove
the
following
assertion:
(∗)
con
Suppose
that
we
have
a
sequence
of
points
{
x
n
},
where
x
n
∈
def
X
L
n
(L
tm
).
Let
x
n
=
ψ
n
(
x
n
)
∈
X
L
(L
tm
).
Then
there
exists
a
subse-
quence
of
{x
n
}
which
converges
p-adically
in
X
L
(M).
47
(Note
that
by
choosing
any
proper
model
over
O
L
of
X
L
,
we
obtain
a
p-adic
topology
on
X
L
(L
tm
)
which
is
easily
seen
to
be
independent
of
the
model
chosen.)
By
replacing
K
be
a
finite
extension
of
K,
we
may
make
the
following
simplifying
assumptions:
(1)
X
K
has
stable
reduction.
We
denote
by
X
→
Spec(O
K
)
the
unique
stable
extension
of
X
K
over
O
K
.
We
denote
by
J
→
Spec(O
K
)
the
unique
semi-abelian
scheme
over
O
K
whose
generic
fiber
is
the
Jacobian
of
X
K
.
(2)
X
K
(K)
is
nonempty.
Note
that
replacing
K
by
a
finite
extension
of
K
does
not
affect
the
validity
of
the
con-
vergence
assertion
(∗)
con
.
Let
us
fix
an
n
≥
1.
Let
us
consider
the
morphism
x
n
:
Spec(L
tm
)
→
X
K
.
This
morphism
induces
a
morphism
β
n
:
Γ
L
tm
→
Π
X
K
→
Π
X
K
/(Δ
X
)
<n>
which
is
well-defined
up
to
conjugation
by
an
element
of
Δ
X
.
Then
it
follows
immediately
from
the
definitions
(and
the
fact
that
x
n
arose
from
an
L
tm
-rational
point
of
X
L
n
)
that:
This
morphism
β
n
is
equal
to
the
composite
of
α|
Γ
L
tm
:
Γ
L
tm
→
Π
X
K
with
the
natural
projection
Π
X
K
→
Π
X
K
/(Δ
X
)
<n>
.
Thus,
in
particular,
β
n
is
independent
of
the
choice
of
x
n
.
Moreover,
by
(∗)
non
,
it
follows
that
by
taking
n
to
be
sufficiently
large,
we
may
assume
that
the
restriction
of
β
n
to
Γ
L
tm
·K
(⊆
Γ
L
tm
)
is
nontrivial.
This
implies,
in
particular,
that
x
n
does
not
factor
through
any
finite
extension
of
K.
Note
that
by
properness,
x
n
extends
to
a
morphism
ξ
n
:
Spec(O
L
tm
)
→
X
.
Let
K
tm
be
the
algebraic
closure
of
K
in
L
tm
.
Thus,
K
tm
is
a
maximal
tamely
ramified
extension
of
K.
Let
us
denote
by
Ω
O
L
tm
/O
K
tm
the
O
L
tm
-module
of
p-adically
continuous
differentials
of
O
L
tm
over
O
K
tm
.
Thus,
Ω
O
L
tm
/O
K
tm
is
a
free
O
L
tm
-module
of
rank
one.
Lemma
9.1.
Ω
O
L
tm
/O
K
tm
.
By
differentiating
ξ
n
,
we
obtain
a
natural
morphism
dξ
n
:
ξ
n
∗
ω
X
/O
K
→
Proof.
Suppose
that
x
n
is
defined
over
some
finite
tamely
ramified
extension
L
of
L.
Suppose,
moreover,
that
L
contains
a
finite
tamely
ramified
extension
K
of
K
such
that
m
K
·
O
L
=
m
L
.
Then,
if
we
equip
Spec(O
K
),
Spec(O
K
)
and
Spec(O
L
)
with
the
log
structures
defined
by
the
special
points,
and
X
with
the
log
structure
whose
monoid
is
the
sheaf
of
functions
invertible
on
the
generic
fiber,
we
obtain
a
log
morphism
48
Spec(O
L
)
log
→
X
log
compatible
with
ξ
n
and
Spec(O
K
)
log
→
Spec(O
K
)
log
.
Moreover,
since
ω
X
/O
K
=
Ω
X
log
/Spec(O
K
)
log
,
and
Ω
O
L
tm
/O
K
tm
=
O
L
tm
⊗
O
L
Ω
Spec(O
L
)
log
/Spec(O
K
)
log
,
we
thus
obtain
dξ
n
by
means
of
the
functoriality
of
logarithmic
differentials.
The
naturality
of
dξ
n
is
easily
checked
by
looking
at
the
generic
fibers.
def
0
Let
Γ
Ω
X
=
H
(X
,
ω
X
/O
K
).
Then
we
propose
to
prove
in
the
present
situation
that
(∗)
ind
There
exists
a
number
n
0
depending
only
on
K
such
that
for
all
n−n
0
n
≥
n
0
,
the
natural
morphism
Γ
Ω
Z
induced
X
→
Ω
O
L
tm
/O
K
tm
⊗
Z/p
by
dξ
n
is
independent
of
x
n
(i.e.,
depends
only
on
α).
To
keep
the
notation
simple,
let
us
note
that
by
replacing
K
by
a
tamely
ramified
extension
of
K,
and
then
replacing
L
by
an
unramified
extension
of
L,
we
may
assume
that
x
n
is,
in
fact,
defined
over
L.
(Note
that
replacing
L
and
K
by
extensions
in
this
fashion
will
not
affect
the
validity
of
(∗)
ind
.)
Thus,
we
shall
regard
x
n
and
ξ
n
as
morphisms
Spec(L)
→
X
K
and
Spec(O
L
)
→
X
,
respectively.
def
Ω
Let
P
=
P(Γ
Ω
X
).
(That
is,
P
is
the
projective
space
over
O
K
defined
by
Γ
X
.)
Then
there
is
(by
the
definition
of
Γ
Ω
X
)
a
natural
finite
morphism
λ
K
:
X
K
→
P
K
.
Let
N
→
Spec(O
K
)
be
the
Néron
model
of
J
X
.
Thus,
J
⊆
N
is
an
open
subscheme
of
N
.
(1)
Since
X
K
(K)
is
nonempty,
we
can
use
a
K-rational
point
of
X
K
to
identify
J
X
with
J
X
.
Thus,
we
obtain
a
morphism
X
K
→
N
K
=
J
X
.
By
the
defining
property
of
the
Néron
model,
and
the
fact
that
O
L
is
geometrically
regular
over
O
K
,
it
follows
that
composing
this
morphism
with
x
n
gives
rise
to
a
morphism
ζ
L
:
Spec(O
L
)
→
N
.
Moreover,
(after
possibly
replacing
K
by
a
finite
unramified
extension
of
K),
we
may
assume
that
there
exists
some
ζ
K
:
Spec(O
K
)
→
N
such
that
ζ
K
and
ζ
L
map
the
special
points
of
Spec(O
K
)
and
Spec(O
L
),
respectively,
to
the
same
geometric
connected
component
of
the
special
fiber
(1)
of
N
.
Thus,
by
translating
by
ζ
K
,
we
may
assume
that
our
identification
of
J
X
with
J
X
is
such
that
ζ
L
maps
into
J
⊆
N
.
We
denote
the
resulting
morphism
by
j
n
:
Spec(O
L
)
→
J
.
Now
let
us
consider
the
Γ
K
-module
H
1
(X
K
,
Z
p
(1))
=
H
1
(J
K
,
Z
p
(1)).
Here,
the
iden-
tification
of
H
1
(X
K
,
Z
p
(1))
with
H
1
(J
K
,
Z
p
(1))
is
the
identification
induced
by
X
K
→
(1)
J
X
∼
=
J
X
.
This
identification
is
the
same
as
the
standard
identification
since
translation
by
K-valued
points
of
J
induces
the
identity
on
H
1
(J
K
,
Z
p
(1)).
Let
H
cb
⊆
H
1
(J
K
,
Z
p
(1))
be
the
maximal
Z
p
-submodule
such
that
Γ
K
acts
on
H
cb
(−1)
through
a
finite,
unram-
ified
quotient.
(Here,
“cb”
stands
for
“combinatorial.”
Note
that
this
inclusion
H
cb
⊆
F
H
1
(J
K
,
Z
p
(1))
is
Cartier-dual
to
the
quotient
H
X
→
H
X
/H
X
(in
the
notation
of
the
dis-
cussion
following
Definition
6.3
in
Section
6).)
By
the
theory
of
[FC],
Chapter
III,
we
know
def
that
C
X
=
H
1
(J
K
,
Z
p
(1))/H
cb
may
be
(Γ
K
-equivariantly)
identified
with
H
1
(G
K
,
Z
p
(1))
for
some
semi-abelian
scheme
G
→
Spec(O
K
)
which
can
be
represented
as
an
extension
0
→T
→G→A→
0
49
of
an
abelian
scheme
A
→
Spec(O
K
)
by
a
torus
T
→
Spec(O
K
).
Moreover,
the
p-adic
completion
G
of
G
is
equal
to
the
p-adic
completion
J
of
J
.
Indeed,
relative
to
the
identification
G
=
J
,
the
identification
C
X
=
H
1
(G
K
,
Z
p
(1))
is
obtained
as
follows:
Given
a
finite
étale
covering
Q
K
→
J
K
,
we
let
Q
be
the
normalization
of
J
in
Q
K
,
so
Q
is
→
J
=
G
finite
over
J
.
Thus,
by
p-adically
completing,
we
obtain
a
formal
morphism
Q
(whose
relative
differentials
are
annihilated
by
a
power
of
p)
which
can
be
algebrized
to
some
finite
R
→
G
(whose
relative
differentials
are
annihilated
by
a
power
of
p).
Thus,
R
K
→
G
K
is
finite
étale.
In
other
words,
the
correspondence
between
coverings
given
by
Q
K
→
R
K
induces
a
morphism
H
1
(J
K
,
Z
p
(1))
→
H
1
(G
K
,
Z
p
(1))
whose
kernel
is
H
cb
,
and
thus
allows
us
to
identify
H
1
(G
K
,
Z
p
(1))
with
C
X
.
Now
by
p-adically
completing
j
n
:
Spec(O
L
)
→
J
,
applying
G
=
J
,
and
then
alge-
brizing,
we
thus
obtain
a
morphism
g
n
:
Spec(O
L
)
→
G.
Moreover,
g
n
induces
a
morphism
on
cohomology
groups
H
1
(g
n
)
:
H
1
(G
K
,
Z
p
(1))
→
H
1
(Γ
L/K
,
Z
p
(1))
Thus,
we
have
a
commutative
diagram
of
Γ
K
-modules:
H
1
(J
K
,
Z
p
(1))
⏐
⏐
H
1
(G
K
,
Z
p
(1))
H
1
(j
n
)
−→
H
1
(g
n
)
−→
H
1
(Γ
L/K
,
Z
p
(1))
⏐
⏐
id
−→
n
H
Ω
L
⊗
(Z/p
Z)
⏐
⏐
id
H
1
(Γ
L/K
,
Z
p
(1))
−→
n
H
Ω
L
⊗
(Z/p
Z)
where
the
vertical
morphism
on
the
left
is
the
natural
projection
discussed
in
the
preceding
paragraph,
and
the
horizontal
morphisms
on
the
right
are
the
natural
ones.
Recall
that
1
H
Ω
L
is,
by
definition,
the
quotient
of
H
(Γ
L/K
,
O
L
(1))
by
its
torsion
submodule.
Now,
by
sorting
through
the
definitions
(cf.
the
discussion
preceding
Lemma
9.1),
it
is
clear
that
the
composite
morphism
on
the
upper
row
is
completely
determined
by
α
(i.e.,
is
independent
of
the
particular
choice
of
x
n
).
Thus,
it
follows
that
the
composite
morphism
on
the
bottom
row
gives
rise
to
a
morphism
n
:
H
1
(1))
→
H
Ω
⊗
Z/p
n
Z
(G
K
,
O
L
K
which
is
completely
determined
by
α.
Moreover,
since
H
Ω
L
is
p-adically
separated
(see
Con-
sequence
(2)
of
[Falt1]
below),
it
follows
(from
(∗)
non
)
that
n
is
nonzero
for
n
sufficiently
large.
Now,
note
that
G
is
a
smooth
O
K
-scheme
with
an
obvious
compactification
G
such
that
G
−
G
is
a
divisor
with
normal
crossings.
(Indeed,
G
is
a
product
of
G
m
-torsors
over
the
O
K
-proper
scheme
A,
so
we
simply
compactify
each
of
these
G
m
-torsors
to
a
P
1
-bundle.)
In
particular,
it
follows
that
we
can
apply
the
theory
of
[Falt1],
in
the
case
of
good
reduction.
50
Remark.
The
argument
of
[Falt1]
in
the
bad
reduction
case
is
known
to
have
gaps,
which
is
one
reason
why
we
took
pains
to
avoid
applying
[Falt1]
to
J
K
.
Another
reason
is
that
in
the
bad
reduction
case,
one
only
has
“Q
p
-results,”
not
integral
results
(say,
over
Z/p
n
Z),
as
we
need
here.
Let
n
0
be
an
integer
such
that
the
p-adic
valuation
of
the
different
of
K
over
Q
p
is
≤
def
def
the
p-adic
valuation
of
p
n
0
.
Let
n
0
=
n
0
+
ord
p
(g!).
Let
n
0
=
n
0
+
2.
The
consequences
of
the
theory
of
[Falt1]
that
we
use
are
as
follows:
∧
(1)
The
Γ
K
-module
H
1
(G
K
,
K
)
is
Hodge-Tate.
Let
us
denote
by
H
Ω
G
the
1
quotient
of
H
(G
K
,
O
K
(1))
by
the
submodule
that
is
contained
in
the
∧
portion
of
H
1
(G
K
,
K
(1))
of
weight
1.
Thus,
H
Ω
G
⊗
Z
p
Q
p
is
of
weight
inv
0.
Let
Ω
G
be
the
O
K
-module
of
invariant
differentials
(over
O
K
)
on
the
group
scheme
G.
Then
there
is
a
natural
Γ
K
-equivariant
morphism
(Theorems
2.4
and
3.1
of
II.
of
[Falt1])
n
0
n
Ω
inv
·
O
K
)/(p
n
0
+n
·
O
K
)}
→
H
Ω
G
⊗
O
K
{(p
G
⊗
Z/p
Z
with
an
inverse
“up
to
a
factor
of
p
n
0
.”
Here
the
point
of
the
n
0
is
to
take
care
of
the
“g!
·
ρ”
of
[Falt1]:
the
2
=
1
+
1
added
to
n
0
is
to
take
1
care
of
the
p−1
≤
1
that
always
arises,
plus
the
additional
e−1
e
<
1
that
arise
from
the
tame
ramification
which
we
allow
(see
[Fo],
Theorem
1
for
more
details
on
the
computation
of
[Falt1]’s
“ρ”).
(2)
H
Ω
L
is
p-adically
separated.
Moreover,
there
is
a
natural
Γ
K
-equivariant
morphism
(Theorem
4.2
of
I.
of
[Falt1])
n
Ω
O
L
/O
K
⊗
O
K
{(p
n
0
·
O
K
)/(p
n
0
+n
·
O
K
)}
→
H
Ω
L
⊗
Z/p
Z
with
an
inverse
“up
to
a
factor
of
p
n
0
.”
In
particular,
by
consid-
ering
weights,
it
follows
that
n
factors
through
the
quotient
H
Ω
G
of
1
H
(G
K
,
O
K
(1)).
(3)
The
morphisms
(and
inverses
up
to
a
factor)
of
(1)
and
(2)
are
com-
patible
with
each
other,
n
,
and
the
natural
morphism
Ω
inv
G
→
Ω
O
L
/O
K
induced
by
differentiating
g
n
.
That
is
to
say,
we
have
a
commutative
diagram:
n
0
·
O
K
)/(p
n
0
+n
·
O
K
)}
Ω
inv
G
⊗
O
K
{(p
⏐
⏐
−→
n
H
Ω
G
⊗
Z/p
Z
⏐
⏐
Ω
O
L
/O
K
⊗
O
K
{(p
n
0
·
O
K
)/(p
n
0
+n
·
O
K
)}
−→
n
H
Ω
L
⊗
Z/p
Z
51
(where
the
horizontal
morphisms
are
the
morphisms
of
(1)
and
(2),
re-
spectively,
the
vertical
morphism
on
the
left
is
induced
by
differentiating
g
n
,
and
the
vertical
morphism
on
the
right
is
induced
by
n
).
These
assertions
allow
us
to
immediately
conclude
(∗)
ind
,
as
follows:
Since
n
is
determined
by
α,
it
thus
follows
that
the
morphism
(induced
by
differentiating
g
n
)
Ω
inv
G
→
Ω
O
L
/O
K
⊗
(Z/p
n−n
0
Z)
is
determined
by
α.
On
the
other
hand,
Ω
inv
can
naturally
be
identified
with
G
inv
inv
Ω
Ω
J
.
Moreover,
Ω
J
can
naturally
be
identified
with
Γ
X
.
This
completes
the
proof
of
(∗)
ind
.
Lemma
9.2.
n
is
nonzero.
Ω
n
For
n
sufficiently
large,
the
morphism
H
Ω
G
→
H
L
⊗
(Z/p
Z)
induced
by
Proof.
This
follows
formally
from
(∗)
non
.
Now
we
are
ready
to
tackle
the
convergence
assertion
(∗)
con
.
Thus,
we
go
back
to
working
over
L
tm
and
K
tm
.
In
particular,
by
composing
x
n
with
λ
K
and
applying
the
valuative
criterion
for
properness,
we
obtain
a
morphism
λ
n
:
Spec(O
L
tm
)
→
P
.
Moreover,
this
O
L
tm
-valued
point
λ
n
of
the
projective
space
P
is
precisely
the
point
defined
by
the
→
Ω
O
L
/O
K
⊗
O
L
O
L
tm
=
Ω
O
L
tm
/O
K
tm
(obtained
by
differentiating
g
n
).
morphism
Ω
inv
G
(Note
that
here
we
use
the
fact
remarked
above
that
Ω
O
L
tm
/O
K
tm
is
a
free
O
L
tm
-module
of
rank
one.)
Thus,
(∗)
ind
and
Lemma
9.2
imply
that
the
points
λ
n
of
P
(O
L
tm
)
converge
L
tm
)
=
P
(M).
On
the
other
hand,
by
“Krasner’s
Lemma”
p-adically
to
a
point
λ
∞
∈
P
(
O
(see,
e.g.,
[Kobl],
p.
70),
since
λ
K
:
X
K
→
P
K
is
finite,
it
follows
that
some
subsequence
{x
n
i
}
of
{x
n
}
converges
to
a
point
x
∞
∈
X
K
(M).
This
proves
the
assertion
(∗)
con
,
as
desired.
In
fact,
we
can
say
more.
For
m
≥
1,
let
x
m
n+m
in
X
L
m
(L
tm
)
under
n
be
the
image
of
x
n+m
the
natural
morphism
X
L
→
X
L
m
.
Thus,
we
obtain
a
sequence
{
x
m
n
}
(in
the
index
n)
of
m
tm
m
points
of
X
L
(L
).
Since
X
L
→
X
L
is
finite,
then
by
appying
“Krasner’s
Lemma”
again,
m
see
that
a
subsequence
of
the
sequence
{
x
m
n
}
converges
p-adically
in
X
L
(M).
Moreover,
by
the
process
of
“Cantor
diagonalization”
(of
elementary
analysis),
we
thus
see
that
we
have
proven
the
following
key
result
(which
is
the
main
result
of
this
Section):
Lemma
9.3.
We
assume
notation
as
in
the
first
paragraph
of
this
Section.
Suppose
that
n
∈
X
L
n
(L
tm
).
Then
there
exists
a
subsequence
we
have
a
sequence
of
points
{
x
n
},
where
x
{
x
n
i
}
of
this
sequence
with
the
following
property:
For
each
m
≥
0,
the
sequence
obtained
m
by
projecting
those
x
n
i
with
n
i
≥
m
to
X
L
m
(M)
converges
to
some
x
m
∞
∈
X
L
(M).
In
particular,
X
L
∞
(M)
is
nonempty.
m
Proof.
The
last
assertion
is
proven
by
noting
that
the
x
m
∞
∈
X
L
(M)
form
a
compatible
∞
system,
hence
define
a
point
of
x
∞
∞
∈
X
L
(M).
52
Finally,
before
proceeding,
we
make
the
following
technically
trivial,
but
important
observation:
Since
the
arithmetic
fundamental
group
of
X
L
∞
is
just
Γ
L
(i.e.,
Im(α
s
)),
it
follows
that
the
morphism
Γ
M
→
Π
X
K
∞
induced
on
fundamental
groups
by
the
image
in
X
K
(M)
of
any
point
(e.g.,
x
∞
∞
)
of
X
L
(M)
is
none
other
than
the
restriction
of
α
:
Γ
L
→
Π
X
K
to
Γ
M
.
Section
10:
Uniqueness
and
Rationality
of
the
Limit
Point
We
continue
with
the
notation
of
the
preceding
Section.
In
this
Section,
we
would
∞
like
to
show
that
the
point
x
∞
∞
∈
X
L
(M)
constructed
in
Lemma
9.3
is
the
unique
element
of
X
L
∞
(M),
and
that
it
in
fact
descends
to
a
point
of
X
L
∞
(L).
We
remark
that
much
of
the
material
of
this
Section
is
not
absolutely
logically
necessary
for
the
proof
of
the
main
results
of
this
paper,
but
is
included
partly
for
aesthetic
reasons,
and
partly
because
its
inclusion
(in
the
opinion
of
the
author)
makes
the
proof
of
the
main
results
of
the
paper
more
transparent.
First,
let
us
make
the
trivial
observation
that
any
point
in
X
L
∞
(M)
arises
as
a
x
∞
∞
(as
n
tm
in
Lemma
9.3)
for
some
sequence
of
x
n
∈
X
L
(L
).
(Indeed,
this
follows
by
using
the
fact
that
M
=
(L
tm
)
∧
to
approximate
the
images
of
the
given
point
in
the
various
X
L
n
(M)’s.)
Thus,
without
loss
of
generality,
we
can
apply
the
results
of
the
discussion
of
Section
9
to
our
analysis
of
an
arbitrary
point
of
X
L
∞
(M).
Now
recall
the
limit
point
λ
∞
∈
P
(M)
of
the
discussion
following
Lemma
9.2.
This
limit
point
is
clearly
independent
of
the
choice
of
the
particular
sequence
{
x
n
}
under
∧
consideration
–
i.e.,
it
depends
only
on
α.
More
precisely,
λ
∞
∈
P
(M)
⊆
P
(L
)
is
the
point
defined
by
the
surjection
∧
inv
Γ
Ω
X
⊗
O
K
L
=
Ω
G
⊗
O
K
L
∧
∧
Ω
=
H
Ω
G
⊗
O
L
→
H
L
⊗
Z
p
Q
p
=
Ω
O
L
/O
K
⊗
O
L
L
∧
K
induced
by
α
(cf.
the
discussion
preceding
Lemma
9.2).
Note
that
Γ
L
acts
naturally
on
all
of
these
modules,
and
that
this
surjection
is
Γ
L
-equivariant.
Indeed,
this
follows
by
transport
of
structure
from
the
fact
that
α
is
defined
on
Γ
L
,
not
just
on
Γ
L
tm
·K
.
Thus,
we
conclude
the
following
∞
Lemma
10.1.
The
image
λ
∞
∈
P
(M)
of
x
∞
∞
∈
X
L
(M)
under
the
natural
morphism
X
L
∞
→
X
K
→
P
K
is
independent
of
x
∞
∞
and,
moreover,
λ
∞
is
defined
over
L.
53
The
next
step
is
to
observe
that
this
argument
can
be
modified
so
as
to
show
that
n
the
image
of
x
∞
∞
in
the
projective
space
defined
by
the
differentials
on
X
L
(for
n
≥
1)
is
independent
of
x
∞
∞
and
defined
over
L.
To
see
this,
we
need
to
introduce
some
new
objects.
First,
let
us
fix
n.
Now
observe
that
(after
possibly
enlarging
K)
there
exists
a
finite
étale
n
→
Spec(K)
of
X
K
with
stable
reduction
Z
n
→
Spec(O
K
),
together
with
a
covering
Z
K
of
L
such
that
finite
Galois
extension
L
n
∼
X
n
=
X
L
n
⊗
L
L
⊗
K
L
=
Z
n
=
Z
K
L
L
tm
),
where
L
tm
is
the
composite
Now
let
m
≥
n,
and
consider
x
m
∈
X
L
m
(L
tm
)
⊆
X
L
m
(
L
(as
extensions
of
L).
(Note
that
L
tm
is
then
a
maximal
tamely
ramified
of
L
tm
and
L
n
Thus,
by
projecting
x
,
we
obtain
a
morphism
extension
of
L.)
m
to
a
point
of
Z
K
n
z
m
:
Spec(O
L
tm
)
→
Z
n
Let
us
assume
that
we
have
chosen
K
large
enough
so
that
Z
K
(K)
is
nonempty,
so
that
(1)
n
n
(i.e.,
the
Jacobian
of
Z
we
can
identify
J
Z
K
n
.
Let
N
Z
n
→
Spec(O
K
)
be
the
K
)
with
J
Z
K
K
n
Néron
model
of
Z
K
over
O
K
.
More
generally,
in
the
following
discussion
we
shall
denote
by
N
(−)
the
Néron
model
of
any
proper
hyperbolic
curve
“(−)”
over
a
discretely
valued
field.
In
fact,
we
shall
even
use
this
notation
for
proper
hyperbolic
curves
over
inductive
limits
of
discretely
valued
fields
in
which
case
this
notation
is
to
be
taken
to
mean
the
corresponding
inductive
limit
of
the
Néron
models.
Also,
we
shall
denote
by
Comp(N
(−)
)
the
(inductive
limit
of)
finite
abelian
(étale)
group
scheme(s)
of
connected
components
of
the
special
fiber
of
the
Néron
model
N
(−)
.
The
main
technical
difficulty
that
we
must
overcome
in
order
to
apply
the
argument
n
of
Section
9
(for
X
K
)
to
the
curves
Z
K
is
the
following:
The
points
that
we
are
interested
(1)
∼
n
tm
n
in
are
the
points
z
m
of
Z
K
(
L
)
=
Z
(O
L
n
=
J
Z
n
,
tm
).
These
points
define
points
of
J
Z
K
K
hence
points
N
n
z
m
:
Spec(O
L
tm
)
→
N
Z
L
tm
Then,
in
order
to
apply
the
argument
of
Section
9,
we
must
show
that:
(After
replacing
K
by
a
finite
extension
of
K
which
is
independent
of
N
m),
z
m
maps
the
special
point
of
Spec(O
L
tm
)
to
the
same
component
tm
n
of
Comp(N
Z
)
as
some
K
-valued
point
of
N
Z
n
.
L
tm
L
tm
54
(see
Lemma
10.2
below
for
an
alternate
formulation.)
To
show
this,
we
reason
as
follows:
in
L
tm
.
Thus,
L
tm
is
a
totally
unr
for
the
maximal
unramified
extension
of
L
Let
us
write
L
tm
is
a
union
of
finite
Galois
extensions
unr
.
In
particular,
L
tamely
ramified
extension
of
L
unr
which
are
of
degree
(over
L
unr
)
prime
to
p.
In
particular,
it
follows
that
the
cokernel
of
L
of
the
inclusion
Comp(N
Z
n
)
→
Comp(N
Z
n
)
L
unr
L
tm
is
annihilated
by
integers
prime
to
p.
(Indeed,
this
follows
by
applying
the
“trace
map”
tm
over
L
unr
)
to
the
second
“Comp,”
and
observing
that
(since
(for
subextensions
of
L
tm
/
L
unr
)
clearly
acts
trivially
on
Comp(N
Z
n
))
this
trace
map
is
just
multiplication
Gal(
L
L
tm
unr
is
an
unramified
extension
of
by
some
integer
prime
to
p.)
On
the
other
hand,
since
L
it
follows
(essentially
from
the
definition
of
the
Néron
model)
that
N
Z
n
=
N
Z
n
.
In
L,
L
unr
L
particular,
since
L
is
a
discretely
valued
field,
it
follows
that
Comp(N
Z
n
)
is
finite.
In
other
L
words,
we
conclude
that:
the
p-torsion
of
Comp(N
Z
n
)
is
annihilated
by
some
(finite)
L
tm
power
of
p.
We
are
now
ready
to
prove
the
following
Lemma:
Lemma
10.2.
After
possibly
enlarging
K
by
a
larger
field
that
is
independent
of
m,
we
J
∼
tm
)
→
J
(1)
may
arrange
that
the
morphism
z
m
:
Spec(
L
n
=
J
Z
n
induced
by
z
m
extends
to
a
Z
K
K
n
)
→
N
,
for
all
m
≥
n.
morphism
Spec(O
L
Z
tm
tm
K
Proof.
Indeed,
without
enlarging
K,
it
follows
from
our
observation
above
concerning
the
J
n
for
some
p-torsion
of
Comp(N
Z
n
)
that
a
·
z
m
extends
to
(z
N
)
m
:
Spec(O
L
tm
)
→
N
Z
K
L
tm
positive
integer
a
whose
order
at
the
prime
p
is
independent
of
m.
Now
let
us
assume
n
annihilated
(by
enlarging
K
–
independently
of
m)
that
all
of
the
p-torsion
points
of
J
Z
K
n
by
a
are
rational
over
K.
Since
all
prime-to-p
torsion
points
of
J
Z
K
are
rational
over
K
tm
,
it
thus
follows
that
the
morphism
“multiplication
by
a”
on
N
Z
n
tm
is
finite
over
a
K
n
⊗
O
tm
⊆
N
n
O
.
Since
(z
N
)
m
neighborhood
of
the
image
of
(z
N
)
m
⊗
O
K
O
K
tm
in
N
Z
K
K
K
Z
K
tm
J
J
was
constructed
by
multiplying
z
m
by
a,
it
thus
follows
that
z
m
extends
to
a
morphism
Spec(O
L
tm
)
→
N
Z
n
tm
,
as
desired.
K
Now
that
we
have
Lemma
10.2,
we
can
use
the
extended
morphism
of
Lemma
10.2
to
construct
the
analogue
of
the
morphism
“g
n
”
of
Section
9
(over
some
tamely
ramified
extension
of
K).
Then
the
rest
of
the
argument
of
Section
9
goes
through
without
difficulty.
More
precisely:
def
def
0
n
n
n
Let
Γ
Ω
=
P(Γ
Ω
Z
n
=
H
(Z
,
ω
Z
n
/O
K
).
Let
P
Z
n
).
(That
is,
P
is
the
projective
space
Ω
Ω
over
O
K
defined
by
Γ
Z
n
.)
Then
there
is
(by
the
definition
of
Γ
Z
n
)
a
natural
finite
morphism
n
n
n
λ
nK
:
Z
K
→
P
K
.
Let
λ
nm
:
Spec(O
L
be
the
morphism
obtained
by
composing
z
m
tm
)
→
P
n
with
λ
K
.
Then,
just
as
in
Section
9,
one
sees
via
p-adic
Hodge
theory
that
as
m
→
∞
(and
55
tm
)
=
P
n
((
L
tm
)
∧
).
Moreover,
n
remains
fixed),
the
λ
nm
converge
to
a
point
in
λ
n
∞
∈
P
n
(
O
L
x
m
}.
Finally,
just
as
we
saw
for
Lemma
10.1,
λ
n
∞
is
independent
of
the
original
sequence
{
by
transport
of
structure
and
the
fact
that
α
(and,
here,
X
L
n
)
are
defined
over
L,
it
follows
n
tm
)
∧
),
where
P
X
n
is
the
projective
space
over
that
if
we
regard
λ
n
∞
as
a
point
of
P
X
((
L
L
associated
to
H
0
(X
L
n
,
ω
X
L
n
/L
)
n
(for
the
original
L),
then
λ
n
∞
∈
P
X
(L).
Thus,
we
see
that
we
obtain
the
following
analogue
of
Lemma
10.1:
∞
∞
Lemma
10.3.
The
image
λ
n
∞
∈
P
X
(M)
of
x
∞
∈
X
L
(M)
under
the
natural
morphism
n
n
X
L
∞
→
X
L
n
→
P
X
is
independent
of
x
∞
∞
and,
moreover,
λ
∞
is
defined
over
L.
n
Now
let
us
recall
some
basic
facts
on
hyperelliptic
curves:
Lemma
10.4.
Let
Q
be
a
proper
hyperbolic
curve
over
an
algebraically
closed
field
Ω
of
characteristic
zero.
Then:
(1)
If
Q
is
hyperelliptic
(i.e.,
admits
a
“g
2
1
,”
or
linear
system
of
dimension
1
and
degree
2),
then
the
g
2
1
is
unique.
(2)
If
Q
is
non-hyperelliptic,
then
the
canonical
morphism
from
Q
into
the
projective
space
associated
to
H
0
(Q,
ω
Q/Ω
)
is
an
embedding.
(3)
Suppose
that
W
→
Q
is
a
finite
étale
covering,
where
W
is
connected.
Then
if
Q
is
non-hyperelliptic,
so
is
W
.
(4)
Suppose
that
W
→
Q
(where
W
is
connected)
is
a
cyclic
étale
covering
of
degree
m
>
2.
Then
W
is
non-hyperelliptic.
Thus,
in
particular,
none
of
the
X
L
n
is
hyperelliptic,
for
n
≥
2.
Proof.
For
proofs
of
assertions
(1)
and
(2),
we
refer
to
[Harts],
Chapter
IV,
§5,
Propo-
sitions
5.2
and
5.3.
As
for
(3),
the
push-forward
of
a
g
2
1
on
W
via
W
→
Q
is
a
g
2
1
on
Q,
so
(3)
follows
immediately.
Now
let
us
consider
assertion
(4).
Let
σ
be
a
generator
of
the
Galois
group
of
W
over
Q.
If
W
admits
a
g
2
1
,
it
is
unique,
hence
stabilized
by
σ.
But
this
means
that
there
exists
some
rational
function
f
W
on
W
in
this
g
2
1
which
satisfies
σ
−1
(f
W
)
=
T
(f
W
),
where
T
is
some
linear
fractional
transformation
with
coefficients
in
Ω.
Note
that
since
σ
has
finite
order,
so
does
T
.
Thus,
if
we
diagonalize
T
(by
choosing
a
different
f
W
),
we
may
assume
that
σ
−1
(f
W
)
=
λ
·
f
W
,
for
some
nonzero
λ
∈
Ω.
(Note
that
T
cannot
be
parabolic
(i.e.,
a
transformation
of
the
form
f
W
→
f
W
+
λ,
for
some
m
nonzero
λ
∈
Ω)
since
T
is
of
finite
order
and
Ω
is
of
characteristic
zero.)
Thus,
f
W
(which
56
is
a
constant
multiple
of
the
norm
(relative
to
W
→
Q)
of
f
W
)
defines
a
rational
function
f
Q
on
Q.
Now
f
Q
is
contained
in
some
g
2
1
on
Q,
so
its
zeroes
have
order
equal
to
1
or
2.
On
the
other
hand,
f
Q
has
an
m
th
root
in
the
function
field
of
W
,
so
it
follows
from
the
assumption
that
m
>
2
that
the
covering
W
→
Q
must
be
ramified
at
the
zeroes
of
f
Q
,
which
is
absurd.
This
contradiction
completes
the
proof.
We
summarize
our
efforts
in
Sections
9
and
10
as
follows:
Corollary
10.5.
Let
M
be
the
p-adic
completion
of
a
maximal
tamely
ramified
extension
L
tm
of
L.
Suppose
that
we
are
given
a
nondegenerate
α
:
Γ
L
→
Π
X
K
,
which
thus
gives
rise
to
X
L
n
,
X
L
∞
.
Suppose
that
for
each
n
≥
0,
X
L
n
(L
tm
)
is
nonempty.
Then
it
follows
that
the
set
X
L
∞
(M)
consists
of
precisely
one
point,
which
is,
in
fact,
contained
in
X
L
∞
(L)
⊆
X
L
∞
(M).
In
particular,
there
exists
a
unique
L-valued
point
of
Spec(L)
→
X
K
whose
induced
morphism
on
fundamental
groups
(for
an
appropriate
choice
of
base-points)
is
the
mor-
phism
α
:
Γ
L
→
Π
X
K
.
∞
Proof.
Indeed,
if
x
∞
if
n
≥
2,
the
image
of
x
∞
∞
∈
X
L
(M),
then
by
Lemma
10.4,
∞
in
n
n
n
∞
X
X
L
(M)
is
determined
by
the
image
λ
∞
of
x
∞
in
P
(M).
But,
by
Lemma
10.3,
λ
n
∞
is
∞
n
independent
of
x
∞
and
defined
over
L.
Thus,
it
follows
that
the
image
of
x
∞
∞
in
X
L
(M)
∞
(for
n
≥
2)
is
independent
of
x
∞
∞
and
defined
over
L.
But,
by
the
definition
of
X
L
,
this
∞
∞
means
that
x
∞
itself
is
“independent
of
x
∞
”
and
defined
over
L.
(In
other
words,
X
L
∞
(M)
consists
of
precisely
one
point,
which
is,
in
fact,
defined
over
L.)
The
last
sentence
is
a
formal
consequence
of
the
rest
of
the
Corollary.
This
completes
the
proof.
Section
11:
Hodge-Tate
Representations
of
Infinite
Rank
Let
K
be
a
finite
extension
of
Q
p
.
Let
X
K
→
Spec(K)
be
a
proper
hyperbolic
curve
over
K.
Let
U
K
be
its
generic
point.
Let
H
U
be
the
abelianization
of
Δ
U
.
Thus,
H
U
is
a
Z
p
-flat
topological
Γ
K
-module
of
infinite
rank.
Note
that
we
have
a
natural
surjection
I
I
H
U
→
H
X
→
H
X
(cf.
the
discussion
following
Definition
6.3
for
the
definition
of
H
X
).
M
Let
H
U
⊆
H
U
be
the
kernel
of
this
surjection.
Note
that
the
inertia
groups
(defined
by
closed
points
of
X
K
)
of
Δ
U
all
map
into
H
U
M
.
Let
H
U
P
⊆
H
U
M
denote
the
closure
of
the
image
of
all
these
inertia
groups.
(Here
the
“P
”
stands
for
“points.”)
Thus,
H
U
/H
U
P
may
be
identified
with
H
X
,
hence
is
of
finite
rank
over
Z
p
.
In
this
Section,
we
would
like
to
analyze
H
U
,
and,
in
particular,
H
U
P
in
greater
detail.
For
simplicity,
let
us
assume
(by
replacing
K
by
a
finite
extension
of
K)
that
X
K
has
stable
reduction
over
O
K
,
and
that
there
exists
a
K-valued
point
x
∈
X
K
(K).
Let
℘
∈
X
K
be
a
closed
point
of
X
K
.
Then
there
are
finitely
many
points
y
1
,
.
.
.
,
y
r
∈
57
X
K
(K)
=
Hom(Spec(K),
X
K
)
that
map
to
℘.
For
i
=
1,
.
.
.
,
r,
let
I
y
i
⊆
Δ
U
denote
“the”
inertia
group
(well-defined
up
to
conjugation
by
an
element
of
Δ
U
)
associated
to
y
i
.
Thus,
projection
to
the
quotient
Δ
U
→
H
U
yields
a
map
def
I
℘
=
r
I
y
i
→
H
U
P
i=1
Note,
moreover,
that
I
℘
has
a
natural
structure
of
Γ
K
-module
(given,
for
instance,
by
conjugating
by
the
image
of
a
section
Γ
K
→
Π
U
K
(cf.
Definition
5.1)
induced
by
x
∈
X
K
(K)
–
note,
however,
that
the
Γ
K
-action
is
independent
of
the
choice
of
x).
Relative
to
this
Γ
K
-action
on
I
℘
,
the
above
morphism
is
Γ
K
-equivariant.
Moreover,
by
letting
℘
range
over
all
closed
points
of
X
K
other
than
that
defined
by
x,
we
obtain
a
continuous
Γ
K
-equivariant
morphism
def
I
℘
→
H
U
P
Ξ
:
M
x
=
℘
=
x
In
fact,
Lemma
11.1.
The
morphism
Ξ
is
an
isomorphism.
Proof.
Note
that
Π
U
K
is
equal
to
the
inverse
limit
of
Π
V
K
,
where
the
limit
is
taken
over
all
open
subsets
V
K
⊆
X
K
−
{x}.
For
such
a
V
K
,
we
can
define
H
V
P
just
as
we
defined
H
U
P
,
and
it
is
well-known
that
H
V
P
is
the
direct
sum
of
the
I
℘
,
where
the
sum
is
taken
over
all
℘
∈
X
K
−
{x}
−
V
K
–
indeed,
this
follows
from
the
well-known
(from
elementary
algebraic
topology)
structure
of
the
homology
group
of
a
Riemann
surface
obtained
by
removing
a
finite
number
of
points
from
a
compact
Riemann
surface.
Passing
to
the
limit
proves
the
Lemma.
Next,
we
would
like
to
define
a
quotient
I
℘
→
I
℘
T
as
follows:
Note
that
if
K
℘
is
the
residue
field
of
X
K
at
℘,
then
the
Γ
K
-module
I
℘
(−1)
(where
the
“(−1)”
is
a
Tate
twist)
may
be
naturally
identified
with
Z
p
[Hom
K
(K
℘
,
K)].
(Here,
Z
p
[a
set]
denotes
the
free
Z
p
-
module
generated
by
the
elements
of
the
set.)
Thus,
there
exists
a
unique
nonzero
quotient
I
℘
(−1)
→
Q
℘
stabilized
by
Γ
K
such
that
Γ
K
acts
trivially
on
Q
℘
.
Moreover,
Q
℘
is
a
free
Z
p
-module
of
rank
one.
Let
I
℘
→
I
℘
T
be
the
quotient
obtained
by
tensoring
I
℘
(−1)
→
Q
℘
with
Z
p
(1).
Moreover,
by
taking
the
product
of
these
quotients,
we
obtain
a
quotient
def
M
x
→
M
x
T
=
I
℘
T
℘
=
x
Let
H
U
P
→
H
U
T
be
the
quotient
corresponding
to
this
quotient
under
the
isomorphism
Ξ.
Thus,
the
Γ
K
-action
on
H
U
T
(−1)
is
trivial.
Moreover,
H
U
P
→
H
U
T
has
the
following
universal
property:
58
Lemma
11.2.
Let
N
be
a
Z
p
-flat
topological
Γ
K
-module
such
that
Γ
K
acts
trivially
on
N
(−1).
Then
any
continuous
Γ
K
-morphism
H
U
P
→
N
factors
through
H
U
T
.
Proof.
This
follows
immediately
from
the
construction
of
H
U
T
.
T
Now
let
us
define
the
subquotient
H
X
of
H
U
as
follows:
First,
let
H
U
F
⊆
H
U
denote
the
F
inverse
image
of
H
X
⊆
H
X
under
the
projection
H
U
→
H
X
(cf.
the
discussion
following
F
T
Definition
6.3
for
the
definition
of
H
X
).
Now
let
H
X
be
the
quotient
of
H
U
F
by
the
kernel
P
T
of
H
U
→
H
U
.
Thus,
we
have
an
exact
sequence
of
topological
Γ
K
-modules
T
F
0
→
H
U
T
→
H
X
→
H
X
→
0
Let
us
take
the
continuous
dual
Hom
cont
Z
p
(−,
Z
p
)
of
this
exact
sequence.
This
gives
us
a
new
exact
sequence
of
topological
Γ
K
-modules
F
T
→
C
X
→
C
U
T
→
0
0
→
C
X
(Here,
one
may
think
of
the
“H’s”
as
standing
for
“homology,”
and
the
“C’s”
as
standing
for
“cohomology.”)
In
particular,
we
have
C
U
T
=
∧
Z
℘
T
℘
=
x
where
Z
℘
T
is
defined
to
be
the
dual
of
I
℘
T
.
If
we
pull-back
this
last
exact
sequence
by
Z
℘
T
⊆
C
U
T
,
we
obtain
exact
sequences
of
topological
Γ
K
-modules
F
→
C
℘
T
→
Z
℘
T
→
0
0
→
C
X
T
is
obtained
by
all
of
which
are
of
finite
rank
over
Z
p
.
Moreover,
it
is
clear
that
that
C
X
summing
these
extensions
(over
℘)
and
then
p-adically
completing.
T
We
would
like
to
show
that
C
X
is
“Hodge-Tate”.
Unfortunately,
typically
the
theory
of
Hodge-Tate
Galois
representations
only
goes
through
for
modules
of
finite
rank
over
Z
p
.
On
the
other
hand,
C
℘
T
is
of
finite
rank
over
Z
p
.
Thus,
our
approach
in
the
following
will
T
be
to
use
the
C
℘
T
to
show
that
C
X
is,
in
some
sense,
like
a
Hodge-Tate
representation.
First,
let
X
℘
be
the
singular
curve
obtained
from
X
K
by
considering
the
subsheaf
of
O
X
K
of
functions
f
such
that
f(x)
=
f(℘).
Let
J
℘
be
the
generalized
Jacobian
associated
to
X
℘
(see,
e.g.,
[Ser1],
Chapitre
I,
§1).
Thus,
J
℘
is
an
extension
of
the
abelian
variety
J
X
by
some
torus.
Moreover,
the
p-adic
Tate
module
T
(J
℘
)
of
J
℘
fits
into
an
exact
sequence
of
Γ
K
-modules
59
0
→
I
℘
→
T
(J
℘
)
→
H
X
→
0
In
particular,
one
sees
immediately
that
there
exists
a
quotient
J
℘
→
J
℘
whose
p-adic
Tate
module
fits
into
the
exact
sequence
0
→
I
℘
T
→
T
p
(J
℘
)
→
H
X
→
0
obtained
by
pushing
forward
the
preceding
exact
sequence
by
I
℘
→
I
℘
T
.
Moreover,
C
℘
T
F
may
be
identified
with
the
dual
of
the
pull-back
of
this
exact
sequence
by
H
X
→
H
X
.
T
∨
1
Alternatively,
C
℘
may
be
identified
with
a
certain
quotient
T
p
(J
℘
)
=
H
(J
℘
,
Z
p
)
→
C
℘
T
F
(whose
kernel
is
the
dual
of
H
X
/H
X
).
Note,
in
particular,
that
J
℘
is
an
extension
of
J
X
by
the
trivial
torus
G
m
.
Thus,
J
℘
extends
to
a
semi-abelian
variety
over
O
K
(cf.
the
paragraph
following
Theorem
2.6
of
Chapter
I
of
[FC]).
Next,
let
us
recall
(from
the
theory
of
[FC],
Chapters
I
through
III)
that
J
X
may
be
constructed
as
a
certain
rigid
analytic
quotient
of
a
semi-abelian
variety
J
X
which
has
good
reduction
(i.e.,
the
dimension
of
the
torus
part
is
constant)
over
O
K
by
some
discrete
group.
The
kernel
of
the
induced
pull-back
map
on
cohomology
H
1
(J
X
,
Z
p
)
→
H
1
(
J
X
,
Z
p
)
F
.
If
we
pull-back
the
extension
J
℘
→
J
X
from
J
X
to
J
X
,
is
precisely
the
dual
of
H
X
/H
X
we
then
obtain
an
extension
J
℘
of
J
X
by
the
trivial
torus
of
dimension
one.
Moreover,
J
℘
has
good
reduction
over
O
K
.
Finally,
the
quotient
H
1
(J
℘
,
Z
p
)
→
H
1
(
J
℘
,
Z
p
)
is
precisely
the
quotient
H
1
(J
℘
,
Z
p
)
→
C
℘
T
referred
to
in
the
preceding
paragraph.
Now,
just
as
in
Section
9,
we
can
apply
the
theory
of
[Falt1]
(in
the
case
of
good
def
reduction)
to
J
℘
.
Let
F
0
=
H
1
(X,
O
X
),
where
X
→
Spec(O
K
)
is
the
stable
extension
of
X
K
over
O
K
.
Let
F
℘
be
the
space
of
invariant
differentials
on
J
℘
,
where
J
℘
→
Spec(O
K
)
is
the
extension
of
J
℘
to
a
semi-abelian
variety
over
O
K
.
Thus,
F
0
and
F
℘
are
both
free
of
finite
rank
over
O
K
.
Then
it
follows
from
the
theory
of
[Falt1]
(specifically,
Theorems
2.4
and
3.1
of
II.
of
[Falt1])
that
-linear
morphism
(∗)
℘
There
exists
a
natural
Γ
K
-equivariant,
O
K
→
(F
0
⊗
O
O
)
⊕
(F
℘
⊗
O
O
(−1))
C
℘
T
⊗
Z
p
O
K
K
K
K
K
which
has
an
inverse
up
to
p
n
,
where
n
is
a
nonnegative
integer
that
is
independent
of
℘.
Note
that
the
fact
that
n
is
independent
of
℘
is
of
profound
importance
in
what
follows.
Moreover,
if,
instead
of
J
℘
,
we
had
worked
with
J
℘
(which,
in
general,
has
“bad
(though
stable)
reduction”),
or
J
℘
(which
has
only
potentially
stable
reduction
–
i.e.,
for
general
℘,
one
needs
to
enlarge
K
in
order
to
get
stable
reduction),
we
would
have
been
unable
to
obtain
this
crucial
“independence
of
℘”
from
the
theory
of
[Falt1].
60
Now
observe
that
F
℘
fits
into
a
natural
exact
sequence
F
→
F
℘
→
Z
℘
→
0
0
→
F
X
def
F
=
H
0
(X,
ω
X/O
K
),
and
Z
℘
is
defined
so
as
to
make
the
above
sequence
exact.
where
F
X
T
Now
let
F
X
be
the
sum
of
these
extensions
over
all
℘;
let
F
U
T
be
the
direct
sum
of
the
Z
℘
over
all
℘.
Thus,
we
have
an
exact
sequence
of
flat
O
K
-modules
F
T
→
F
X
→
F
U
T
→
0
0
→
F
X
Thus,
by
taking
the
p-adic
completion
of
the
sum
of
the
morphisms
in
(∗)
℘
,
and
using
the
facts
that:
(i)
the
“n”
of
(∗)
℘
is
independent
of
℘;
T
T
(ii)
both
C
X
and
F
X
are
sums
of
extensions
which
are
of
finite
rank;
we
obtain
the
following
result:
-linear
morphism
of
exact
sequences
of
Proposition
11.3.
We
have
a
natural,
O
K
topological
Γ
K
-modules
0
−→
F
⊗
Z
p
O
C
X
K
⏐
⏐
−→
T
⊗
Z
p
O
C
X
K
⏐
⏐
0
−→
F
(−1))
⊕
(F
0
⊗
)
O
K
O
(F
X
⊗
O
K
O
K
K
−→
T
(−1))
⊕
(F
0
⊗
)
O
K
O
⊗
O
K
O
(F
X
K
K
−→
Z
p
O
C
U
T
⊗
K
⏐
⏐
−→
0
−→
(−1))
O
K
O
(F
U
T
⊗
K
−→
0
which
become
isomorphisms
when
tensored
with
Q
p
.
T
Next,
we
would
like
to
relate
F
X
to
a
certain
space
of
differentials
on
X,
as
follows:
def
Let
E
be
a
finite
set
of
closed
points
of
X
K
.
Let
S
=
Spec(O
K
),
and
let
us
endow
S
with
the
log
structure
defined
by
the
monoid
O
S
−
{0};
denote
the
resulting
log
scheme
by
S
log
.
Let
X
log
[E]
→
S
log
be
the
pointed
stable
log-curve
(see,
e.g.,
[Mzk3],
Section
3,
for
more
details
on
this
terminology)
extending
X
K
⊗
K
K,
equipped
with
all
the
points
61
def
of
X
K
(K)
that
map
to
primes
in
E
as
marked
points.
Let
L[E]
=
ω
X
log
[E]/S
log
.
Then
if
E
⊆
E
,
we
get
a
natural
morphism
X
log
[E
]
→
X
log
[E],
which
lies
under
a
morphism
of
sheaves
L[E]
→
L[E
].
Thus,
by
taking
the
projective
limit
of
the
(X
log
[E],
L[E])
as
E
ranges
over
all
finite
sets
of
closed
points
of
X
K
,
we
obtain
(X
log
[∞],
L[∞]).
Moreover,
note
that
H
0
(X[∞],
L[∞])
is
a
flat,
p-adically
separated
O
K
-module.
If
E
contains
x
and
℘,
then
it
is
easy
to
see
that
we
get
a
natural
inclusion
F
℘
→
H
(X[E],
L[E]),
which
extends
to
a
morphism
F
℘
⊗
O
K
O
K
→
H
0
(X[E],
L[E])
whose
cokernel
is
torsion
free.
Taking
the
limit
with
respect
to
E,
we
thus
obtain
a
map
F
℘
⊗
O
K
O
K
→
H
0
(X[∞],
L[∞]).
In
fact,
it
is
not
difficult
to
see
that,
by
summing
over
the
F
℘
for
all
℘
=
x,
we
get
a
morphism
0
def
T
F
X
⊗
O
K
O
K
→
F
∞
=
H
0
(X[∞],
L[∞])
whose
cokernel
is
torsion
free.
Moreover,
this
morphism
remains
injective
after
p-adic
completion.
Thus,
we
see
from
Proposition
11.3
that
we
get
a
morphism
T
→
F
∧
(−1)
⊗
Z
p
O
Φ
:
C
X
∞
K
(where
the
“∧”
denotes
p-adic
completion).
Moreover,
the
kernel
of
Φ
is
of
weight
zero.
def
Finally,
if
N
is
a
positive
integer,
let
G
∞
[N
]
=
H
0
(X[∞],
L[∞]
⊗N
).
Thus,
G
∞
[N
]
is
a
free
O
K
-module
of
infinite
rank
(such
that
G
∞
[1]
=
F
∞
),
and
we
have
a
natural
multiplication
morphism
⊗
N
O
K
F
∞
→
G
∞
[N
]
Section
12:
The
Preservation
of
Relations
The
purpose
of
this
Section
is
to
show
how
the
material
of
Section
11
will
be
applied
in
Section
13
to
show
the
“preservation
of
relations.”
We
maintain
the
notation
of
Section
11.
Moreover,
let
us
assume
that
we
have
been
given
a
continuous
open
homomorphism
over
Γ
K
θ
:
Π
U
K
→
Π
Y
K
(where
Y
K
→
Spec(K)
is
a
proper
hyperbolic
curve
over
K).
For
simplicity,
let
us
assume
(by
enlarging
K)
that
Y
K
extends
to
a
stable
curve
Y
→
Spec(O
K
)
over
O
K
,
and
that
the
Γ
K
-action
on
H
Y
M
(−1)
(cf.
the
discussion
following
Definition
6.3
for
an
explanation
of
the
notation
“H
M
”)
is
trivial.
Next,
observe
that
θ
induces
a
continuous
morphism
of
Γ
K
-modules
H
U
→
H
Y
whose
cokernel
is
torsion.
Note
that
this
morphism
induces
62
a
map
H
U
F
→
H
Y
F
(cf.
the
discussions
following
Definition
6.3
and
Lemma
11.2
for
an
explanation
of
the
notation
“H
F
”)
whose
cokernel
is
torsion.
Moreover,
the
restriction
of
this
morphism
to
H
U
P
(cf.
the
discussion
at
the
beginning
of
Section
11
for
an
explanation
of
the
notation
“H
U
P
”)
maps
into
H
Y
M
,
hence
(by
Lemma
11.2
and
the
fact
that
Γ
K
acts
T
trivially
on
H
Y
M
(−1)),
we
get
a
morphism
H
X
→
H
Y
F
whose
cokernel
is
torsion.
Taking
T
the
dual
of
this
morphism
yields
an
injection
C
Y
F
→
C
X
(where
C
Y
F
is
the
dual
of
H
Y
F
).
Topologically
tensoring
with
O
K
,
dividing
out
by
the
“weight
zero
part,”
and
applying
the
morphism
Φ
considered
at
the
end
of
Section
11,
we
thus
obtain
(after
further
tensoring
with
Z
p
(1))
an
injection
→
F
∧
H
0
(Y,
ω
Y
/O
K
)
⊗
O
K
O
∞
K
In
this
Section,
we
shall
be
concerned
with
the
issue
of
whether
or
not
this
morphism
“preserves
relations.”
By
this,
we
mean
the
following:
Let
N
be
a
positive
integer.
By
taking
the
N
th
tensor
power
of
this
morphism,
and
then
composing
with
the
completion
of
the
multiplication
morphism
⊗
N
F
∞
→
G
∞
[N
]
considered
at
the
end
of
Section
11,
we
obtain
a
morphism
)
→
(G
∞
[N
])
∧
Ψ
N
:
⊗
N
(H
0
(Y,
ω
Y
/O
K
)
⊗
O
K
O
K
We
would
like
to
know
whether
or
not
Ψ
N
annihilates
the
elements
of
R
N
=
Ker(⊗
N
(H
0
(Y,
ω
Y
/O
K
))
→
H
0
(Y,
ω
Y
⊗N
/O
K
))
def
i.e.,
the
“relations.”
In
the
following,
we
would
like
to
state
a
certain
assertion
that
will
be
proven
in
the
next
Section,
and
explain
why
this
assertion
implies
that
Ψ
N
“preserves
relations.”
First,
let
us
introduce
the
notation
necessary
to
state
the
assertion.
Suppose
that
E
is
a
finite
set
of
points
of
X
K
(K).
Then
the
pointed
stable
curve
X
log
[E]
→
Spec(O
K
)
introduced
at
the
end
of
Section
11
is
defined
over
K,
i.e.,
we
have
X
log
[E]
O
K
→
Spec(O
K
).
Let
p
∈
X[E]
O
K
be
a
generic
point
of
the
special
fiber
of
X[E]
O
K
.
Let
Spf(O
L
)
be
the
completion
of
the
localization
of
X[E]
O
K
at
p.
Thus,
L
(the
quotient
field
of
O
L
)
is
a
p-adic
field
whose
residue
field
is
k(p)
(i.e.,
the
residue
field
of
X[E]
O
K
at
the
prime
p),
and
we
have
a
canonical
L-valued
point
ξ
L
∈
X
K
(L).
Let
Ω
L/K
denote
the
module
of
p-adically
continuous
differentials
of
L
over
K.
Thus,
Ω
L/K
is
an
L-vector
space
of
dimension
one.
Moreover,
note
that
we
have
a
natural
restriction
morphism
(induced
by
ξ
L
)
∧
K
K
→
Ω
L/K
⊗
F
∞
Thus,
if
we
compose
this
morphism
with
the
morphism
→
F
∧
H
0
(Y,
ω
Y
/O
K
)
⊗
O
K
O
∞
K
63
considered
above,
we
get
a
morphism
→
Ω
L/K
⊗
K
K
κ
L
:
H
0
(Y,
ω
Y
/O
K
)
⊗
O
K
O
K
Next,
let
us
observe
that
the
coverings
of
X
K
defined
by
subgroups
of
Π
U
K
are
étale
at
the
point
ξ
L
.
Thus,
it
follows
that
ξ
L
induces
a
natural
morphism
α
L
U
:
Γ
L
→
Π
U
K
(well-defined
up
to
conjugation
by
an
element
of
Δ
U
)
whose
composite
with
Π
U
K
→
Γ
K
L
is
the
natural
morphism
Γ
L
→
Γ
K
.
Let
α
L
Y
:
Γ
L
→
Π
Y
K
be
the
composite
of
α
U
with
θ.
Then
the
assertion
that
will
be
proven
in
Section
13
is
the
following:
(∗)
L−pt
Suppose
that
p
satisfies
the
condition
that
κ
L
is
not
identically
zero.
Then
it
follows
that
there
exists
a
point
(ξ
)
L
∈
Y
K
(L)
such
that
L
α
L
Y
arises
from
(ξ
)
.
L
Note
in
particular,
that
if
α
L
Y
arises
from
(ξ
)
,
then
it
follows
immediately
that
the
group-
theoretically
constructed
morphism
κ
L
coincides
(cf.
the
application
of
the
theory
of
[Falt1]
discussed
in
Section
9)
with
the
natural
restriction
map
on
differentials
induced
by
(ξ
)
L
.
Thus,
if
ρ
∈
R
N
is
a
relation,
then
it
follows
that
the
restriction
of
Ψ
N
(ρ)
∈
(G
∞
[N
])
∧
to
Ω
⊗N
L/K
⊗
K
K
is
zero.
This
argument
already
leads
one
to
believe
that
there
should
be
some
sort
of
connection
between
(∗)
L−pt
and
the
“preservation
of
relations.”
In
fact,
we
have
the
following:
Proposition
12.1.
Suppose
that
(∗)
L−pt
always
holds
(i.e.,
for
all
data
of
the
sort
discussed
above).
Then
Ψ
N
(R
N
)
=
0,
for
all
positive
integers
N
.
def
Proof.
Suppose
that
ρ
∈
R
N
is
such
that
ψ
=
Ψ
N
(ρ)
=
0.
By
dividing
ψ
by
a
suitable
element
of
O
K
,
we
obtain
an
element
ψ
∈
(G
∞
[N
])
∧
such
that
ψ
≡
0
(mod
m
K
)
(where
m
K
⊆
O
K
is
the
maximal
ideal).
Let
us
write
k
for
the
residue
field
O
K
/m
K
.
Write
ψ
for
ψ
considered
modulo
m
K
.
Then
ψ
is
a
section
of
L[E]
⊗N
⊗
k
for
some
E.
By
enlarging
K,
we
may
assume
that
E
consists
solely
of
K-valued
points
of
X
K
.
Moreover,
since
ψ
is
nonzero,
there
exists
some
irreducible
component
of
the
special
fiber
of
X[E]⊗k
over
which
ψ
is
nonzero.
If
we
choose
p
(in
the
above
discussion)
to
be
such
that
p
⊗
O
K
O
K
is
this
irreducible
component,
then
it
follows
that
the
restriction
of
ψ
∈
(G
∞
[N
])
∧
to
Ω
⊗N
L/K
⊗
K
K
will
be
nonzero.
Now
I
claim
that
p
satisfies
the
condition
that
κ
L
is
not
identically
zero:
Indeed,
if
κ
L
were
identically
zero,
then
it
would
follow
that
κ
⊗N
would
be
identically
L
⊗N
zero.
But
the
restriction
of
ψ
to
Ω
⊗
K
K
(which
is
assumed
to
be
nonzero)
is
a
nonzero
L/K
⊗N
K-multiple
of
κ
L
(ρ)
(which
would
have
to
be
zero).
This
contradiction
proves
the
claim.
Thus,
we
are
in
a
position
to
apply
(∗)
L−pt
.
As
discussed
in
the
paragraph
preceding
this
Proposition,
it
then
follows
that
the
restriction
of
ψ
to
Ω
⊗N
L/K
⊗
K
K
is
zero.
Thus,
we
get
a
contradiction.
This
completes
the
proof
of
the
Proposition.
64
Remark.
In
[Mzk2],
where
one
only
considers
isomorphisms
of
π
1
’s,
as
opposed
to
homo-
morphisms
as
we
do
here,
there
is
no
need
to
place
(as
we
did
in
(∗)
L−pt
)
the
condition
on
p
that
κ
L
be
not
identically
zero.
Because
in
the
present
context
it
is
necessary
to
include
such
a
condition
on
p
in
(∗)
L−pt
,
the
author
at
first
did
not
see
how
it
would
be
possible
to
prove
the
“preservation
of
relations”
in
the
present
context.
However,
in
fact,
in
order
to
prove
the
preservation
of
relations
(Proposition
12.1),
it
suffices
to
consider
only
p
for
which
one
knows
already
that
κ
L
is
not
identically
zero.
This
observation
arose
in
discussions
between
the
author
and
A.
Tamagawa.
Section
13:
The
Preservation
of
L-Points
The
purpose
of
this
Section
is
to
verify
the
assertion
(∗)
L−pt
discussed
in
Section
12.
The
technique
is
similar
to
that
employed
in
[Mzk2]
(although
we
do
not
assume
any
knowledge
of
[Mzk2]
in
the
following
discussion).
We
continue
with
the
notation
of
the
preceding
Section.
In
particular,
we
assume
that
we
have
been
given
a
continuous
open
homomorphism
over
Γ
K
θ
:
Π
U
K
→
Π
Y
K
Moreover,
we
assume
that
X
K
extends
to
a
stable
curve
X
→
Spec(O
K
).
Let
p
be
as
in
→
X
(that
is,
X
is
what
we
denoted
by
X[E]
O
(∗)
L−pt
.
Thus,
we
have
a
blow-up
X
K
→
Spec(O
K
).
Choose
a
in
Section
12),
and
p
is
a
generic
point
of
the
special
fiber
of
X
smooth,
affine,
geometrically
connected
open
neighborhood
W
of
p
in
the
special
fiber
of
Let
T
be
the
affine
scheme
whose
coordinate
ring
R
T
is
such
that
Spf(R
T
)
(where
we
X.
along
W
.
Thus,
T
→
Spec(O
K
)
equip
R
T
with
the
p-adic
topology)
is
the
completion
of
X
is
(p-adically)
formally
smooth,
and
T
⊗
k
=
W
.
By
abuse
of
notation,
we
shall
write
p
for
Let
us
write
η
T
for
the
generic
the
prime
of
T
that
maps
to
the
original
p
under
T
→
X.
point
of
T
(regarded
as
a
scheme).
Let
L
be
the
quotient
field
of
the
p-adic
completion
of
(R
T
)
p
.
Thus,
L
is
a
p-adic
field
whose
residue
field
is
k(W
)
(the
function
field
of
W
).
η
T
T
L
Note
that
we
have
natural
morphisms
ξ
X
:
T
→
X;
ξ
X
:
η
T
→
X
K
;
ξ
X
:
Spec(L)
→
η
T
L
X
K
;
ξ
U
:
η
T
→
U
K
;
ξ
U
:
Spec(L)
→
U
K
.
Let
α
η
U
T
:
Γ
η
T
→
Π
U
K
be
the
morphism
η
T
determined
(up
to
conjugation
by
an
element
of
Δ
U
)
by
ξ
U
.
Similarly,
we
have
α
L
U
:
η
T
η
T
L
L
Γ
L
→
Π
U
K
.
Composing
α
U
,
α
U
with
θ
gives
α
Y
:
Γ
η
T
→
Π
Y
K
,
α
Y
:
Γ
L
→
Π
Y
K
.
Now
let
Δ
Y
⊆
Δ
Y
be
a
p-derivate
(see
Definition
0.2)
of
Δ
Y
.
Thus,
Δ
Y
is
an
open,
characteristic
subgroup
of
Δ
Y
.
Note
that
α
η
Y
T
defines
a
section
a
η
Y
T
:
Γ
η
T
→
Π
Y
η
of
T
Π
Y
η
→
Γ
η
T
.
Write
Π
Y
η
for
the
open
subgroup
Im(a
η
Y
T
)
·
Δ
Y
⊆
Π
Y
η
.
Thus,
Π
Y
η
gives
T
T
T
T
rise
to
a
finite
étale
covering
Y
η
T
→
Y
η
T
65
Moreover,
there
exists
some
finite
étale
covering
η
T
→
η
T
such
that
Y
Y
η
T
×
η
T
η
T
∼
×
K
η
T
=
Z
K
Y
for
some
curve
Z
K
→
Spec(K
),
where
K
is
a
finite
extension
of
K
contained
in
K(η
T
).
Let
T
be
the
normalization
of
T
in
η
T
.
Then
T
→
T
is
finite
(since
η
T
is
of
characteristic
zero)
and
generically
étale.
By
means
of
θ,
we
can
pull-back
the
above
covering
to
U
η
T
:
Thus,
we
obtain
an
open
subgroup
Π
U
η
⊆
Π
U
η
,
which
corresponds
to
some
finite
étale
covering
U
η
T
→
U
η
T
T
T
(which
is
geometrically
connected
over
η
T
since
Π
U
η
contains
the
graph
of
α
η
U
T
in
Π
U
η
=
T
T
Γ
η
T
×
Γ
K
Π
U
K
).
This
covering
extends
to
a
finite,
possibly
ramified
covering
X
η
T
→
X
η
T
.
Moreover,
we
have
an
isomorphism
X
X
η
T
×
η
T
η
T
∼
×
K
η
T
=
Z
K
X
for
some
curve
Z
K
→
Spec(K
).
By
enlarging
K
,
we
may
assume
that
this
curve
has
a
stable
extension
Z
X
→
Spec(O
K
).
η
T
:
η
T
→
U
K
.
Now
it
follows
Next,
let
us
go
back
to
considering
the
morphism
ξ
U
η
T
η
T
)
tautologically
from
the
way
we
defined
α
Y
(i.e.,
the
fact
that
it
comes
originally
from
ξ
U
η
T
η
T
η
T
that
ξ
U
lifts
naturally
to
a
point
ξ
U
:
η
T
→
U
η
T
.
Let
ξ
X
:
η
T
→
X
η
T
be
the
result
of
composing
this
morphism
with
U
η
T
→
X
η
T
.
Moreover,
by
restricting
to
η
T
,
we
get
a
point
η
η
T
X
X
ξ
X
T
:
η
T
→
X
η
T
.
Projecting
to
Z
K
,
we
thus
get
a
point
ξ
X
:
η
T
→
Z
K
.
Moreover,
it
Z
T
→
Z
X
,
where
T
⊆
T
is
easy
to
see
that
this
morphism
extends
to
a
morphism
ξ
Z
X
:
T
is
an
open
subscheme
obtained
as
the
“D(f)”
(i.e.,
the
complement
of
V
(f)
⊆
T
)
for
some
f
∈
Γ(T
,
O
T
)
with
the
property
that
f
is
nonzero
at
every
generic
point
of
the
special
fiber
of
T
→
Spec(O
K
).
Indeed,
this
follows
from
the
following
two
facts:
(i)
we
η
T
X
already
have
an
extension
to
T
K
,
since
Z
K
→
X
K
is
finite,
and
ξ
X
:
η
T
→
X
K
extends
to
T
K
;
(ii)
to
extend
from
T
K
to
some
T
,
it
suffices
to
apply
the
valuative
criterion
for
properness
(since
Z
X
→
Spec(O
K
)
is
proper).
Next,
let
us
consider
the
morphism
T
⊆
T
→
T
.
Clearly
(after
possibly
enlarging
K
)
there
exists
a
closed
point
t
∈
T
such
that
if
we
let
S
be
the
completion
of
T
at
t,
then
the
morphism
S
→
S
obtained
by
base-changing
T
→
T
by
S
→
T
has
the
following
properties:
(i)
S
→
S
is
finite
and
flat;
(ii)
S
is
a
finite
disjoint
sum
of
connected
components
which
are
geometrically
irreducible
over
O
K
;
(iii)
each
of
these
components
admits
a
section
over
O
K
.
Let
S
be
any
one
of
these
connected
components.
Then
we
have
the
following:
Lemma
13.1.
The
scheme
S
is
normal.
Proof.
(Note
that
this
is
not
entirely
obvious
since
it
is
not
clear
that
T
is
excellent.)
First,
I
claim
that
the
(flat)
morphism
S
→
T
is
geometrically
regular.
Indeed,
over
the
generic
66
fiber,
this
follows
from
the
fact
that
S
is
regular
and
generically
of
characteristic
zero;
over
a
closed
point
of
T
K
or
T
,
either
the
fiber
of
the
morphism
S
→
T
is
(schematically)
isomorphic
to
the
given
closed
point,
or
it
is
empty.
Thus,
it
remains
to
check
what
happens
over
the
height
one
prime
of
T
which
is
the
special
fiber
of
T
→
Spec(O
K
).
But
over
this
prime,
the
geometric
regularity
of
S
→
T
follows
from
the
excellence
of
T
⊗
k
=
W
(which
is
finitely
generated
over
a
finite
field).
This
completes
the
proof
of
the
claim.
Thus,
it
follows
that
S
→
T
is
geometrically
regular.
Moreover,
T
is
normal;
hence,
we
obtain
that
S
is
normal,
as
desired.
Thus,
S
→
S
satisfies
the
properties
listed
in
Definition
6.3.
Moreover,
if
we
apply
the
base-change
S
→
T
to
all
the
objects
in
the
above
discussion,
it
is
easy
to
see
that
we
are
in
the
situation
discussed
in
Proposition
7.4.
Note,
in
particular,
that
the
existence
of
the
T
morphism
ξ
Z
→
Z
X
shows
that
the
condition
(∗)
S
(cf.
the
discussion
preceding
X
:
T
Definition
6.4)
is
satisfied
by
α
η
X
S
:
Γ
η
S
→
Π
X
η
.
In
fact,
we
also
have
the
following:
S
Lemma
13.2.
α
η
X
S
is
F
-geometric,
hence
also
F
I-geometric.
T
X
Proof.
Indeed,
first
observe
that
restricting
ξ
Z
.
X
gives
rise
to
a
morphism
ξ
1
:
S
→
Z
Since
S
was
constructed
so
as
to
admit
a
section
over
O
K
,
let
σ
:
Spec(O
K
)
→
S
be
such
a
section.
Then
we
may
form
the
composite
ξ
σ
:
S
→
Z
X
of
the
structure
morphism
S
→
Spec(O
K
)
with
ξ
1
◦
σ.
Now
observe
that
ξ
1
and
ξ
σ
both
define
sections
def
of
Z
S
X
=
Z
X
×
O
K
S
→
S
(which
coincide
over
Im(σ)).
Let
D
1
,
D
σ
⊆
Z
S
X
be
the
Weil
divisors
which
are
the
images
of
these
two
sections.
Next,
let
us
observe
that
by
Lemma
13.3
below,
these
two
Weil
divisors
are
Q-Cartier.
Hence
it
follows
that
there
exists
a
positive
integer
N
such
that
N
·
D
1
and
N
·
D
σ
are,
in
fact,
Cartier
divisors,
so
we
may
def
form
the
line
bundle
L
=
O
Z
X
(N
·
D
1
−
N
·
D
σ
).
Note
that
the
line
bundle
L
is
trivial
over
S
Im(σ)
⊆
S
,
hence
over
the
closed
point
of
S
.
Let
J
Z
X
→
Spec(O
K
)
be
the
unique
semi-
X
abelian
scheme
whose
generic
fiber
is
the
Jacobian
of
Z
K
.
Then
it
follows
that
L
defines
an
S
-valued
point
of
J
Z
X
which
arises
from
an
S
-valued
point
of
the
formal
completion
of
J
Z
X
at
the
identity.
But,
sorting
through
the
definitions
(in
particular,
Definitions
6.2,
6.4)
reveals
that
this
implies
that
α
η
X
S
is
F
-geometric,
hence
also
F
I-geometric
(cf.
the
paragraph
following
Defintion
6.4).
def
Lemma
13.3.
The
scheme
Z
S
X
=
Z
X
×
O
K
S
is
normal,
and,
moreover,
any
Weil
X
divisor
D
⊆
Z
S
arising
as
the
graph
of
an
O
K
-morphism
ξ
D
:
S
→
Z
X
is
Q-Cartier.
Proof.
By
Lemma
13.1
above,
S
is
normal.
Moreover,
since
Z
S
X
is
a
generically
smooth
stable
curve
over
S
,
one
checks
easily
that
the
conditions
“R
1
”
and
“S
2
”
of
Serre
(see,
e.g.,
[Mats],
Chapter
7,
§17.I,
Theorem
39)
are
satisfied
by
Z
S
X
,
so
Z
S
X
is
normal.
Now
recall
that
to
say
that
a
Weil
divisor
is
“Q-Cartier”
simply
means
that
some
nonzero
multiple
of
that
Weil
divisor
is
Cartier.
Next,
let
us
observe
that
since
the
graph
of
ξ
D
is
defined
by
a
67
single
equation
in
a
neighborhood
of
any
point
of
the
smooth
locus
of
Z
X
→
Spec(O
K
),
it
suffices
to
check
that
D
is
Q-Cartier
in
a
neighborhood
of
the
nodes
of
the
special
fiber
of
Z
X
→
Spec(O
K
).
In
fact,
it
even
suffices
to
check
that
D
is
Q-Cartier
over
the
spectrum
of
the
completion
of
the
local
ring
of
Z
S
X
at
such
a
node.
Thus,
let
us
write
A
=
Γ(S
,
O
S
);
def
def
B
=
O
K
[[x,
y]]/(xy
−
π
n
);
def
C
=
A[[x,
y]]/(xy
−
π
n
)
where
π
∈
O
K
is
a
uniformizer,
and
n
is
a
positive
integer.
Here,
we
think
of
B
(respec-
tively,
C)
as
the
result
of
completing
Z
X
(respectively,
Z
S
X
)
at
a
node
(respectively,
at
the
inverse
image
of
this
node
via
the
projection
Z
S
X
→
Z
X
).
Recall
that
S
is
affine,
so
S
=
Spec(A).
Now
ξ
D
is
given
by
some
morphism
B
→
A.
Write
s
x
,
s
y
∈
A
for
the
images
of
x
and
y,
respectively,
under
this
morphism.
Note
that
s
x
·
s
y
=
π
n
∈
A.
Moreover,
the
restriction
D
C
of
the
subscheme
D
to
Spec(C)
is
defined
by
the
equations
x
−
s
x
,
y
−
s
y
,
i.e.,
D
C
=
V
(x
−
s
x
,
y
−
s
y
).
Now
let
us
consider
the
regular
function
x
−
s
x
on
Spec(C).
I
claim
that
the
ideals
(x
−
s
x
)
and
(x
−
s
x
,
y
−
s
y
)
in
C
coincide
in
C[
π
1
].
Indeed,
this
follows
from
the
following
equation:
y
−
s
y
=
−y
·
s
y
1
(x
−
s
x
)
∈
C[
]
n
π
π
Thus,
we
conclude
that
on
Spec(C),
the
Weil
divisor
D
C
is
linearly
equivalent
to
a
Weil
divisor
E
on
Spec(C)
which
is
supported
on
the
special
fiber
F
C
of
Spec(C)
→
Spec(O
K
).
On
the
other
hand,
the
fact
that
S
→
Spec(O
K
)
is
geometrically
irreducible
implies
that
(F
C
)
red
has
precisely
two
irreducible
components,
namely,
V
(x)
red
and
V
(y)
red
.
Thus,
it
follows
that
any
divisor
supported
on
F
C
is
Q-Cartier,
as
desired.
This
completes
the
proof
of
the
Lemma.
Thus,
(by
Lemma
13.2)
we
may
apply
Proposition
7.4
to
conclude
that
Y
η
S
admits
a
line
bundle
of
degree
prime
to
p.
Moreover,
by
Proposition
8.1
and
Lemma
8.2,
it
thus
follows
that:
def
Y
L
=
Y
η
T
×
η
T
Spec(L)
has
a
rational
point
over
some
tamely
ramified
extension
L
of
L.
It
is
this
key
result
that
will
allow
us
to
conclude
the
proof
of
(∗)
L−pt
.
Let
us
review
what
we
have
done
so
far,
from
the
point
of
view
of
objects
over
Y
L
.
L
First,
we
have
a
section
a
L
Y
:
Γ
L
→
Π
Y
L
of
Π
Y
L
→
Γ
L
(defined
by
α
Y
).
Moreover,
given
def
any
p-derivate
Δ
Y
⊆
Δ
Y
,
we
obtain
a
finite
étale
covering
Y
L
→
Y
L
defined
by
Π
Y
=
L
L
Im(a
L
Y
)
·
Δ
Y
⊆
Π
Y
L
.
Under
these
circumstances,
we
just
showed
that
Y
L
necessarily
has
L
68
a
rational
point
over
some
tamely
ramified
extension
of
L.
Moreover,
tracing
through
the
definitions,
it
is
easy
to
see
that
the
assumption
placed
on
p
in
(∗)
L−pt
that
κ
L
be
not
identically
zero
means
precisely
(in
the
language
introduced
at
the
beginning
of
Section
9)
that
a
L
Y
is
nondegenerate.
It
thus
follows
that
we
can
apply
Corollary
10.5
to
conclude
that:
L
a
L
Y
:
Γ
L
→
Π
Y
L
arises
from
some
geometric
point
(ξ
)
∈
Y
L
(L).
In
other
words,
the
proof
of
(∗)
L−pt
has
been
completed.
Thus,
by
Proposition
12.1,
we
conclude
the
following:
Corollary
13.4.
Let
θ
:
Π
U
K
→
Π
Y
K
be
a
continuous
open
homomorphism
over
Γ
K
.
Then
θ
“preserves
relations,”
i.e.,
(in
the
notation
of
Section
12),
we
have
Ψ
N
(R
N
)
=
0,
for
all
positive
integers
N
.
Note
that
in
[Mzk2]
(where
we
essentially
dealt
with
the
case
where
θ
is
an
isomor-
phism),
the
preservation
of
relations
is
already
enough
to
conclude
the
proof
of
the
main
theorem.
In
the
present
context,
however,
because
of
the
fact
that
H
U
is
of
infinite
rank,
it
is
necessary
to
go
through
one
more
intermediate
technical
step
before
we
can
complete
the
proof
of
the
(first
part
of
the)
first
main
theorem.
This
step
essentially
amounts
to
showing
that
any
θ
as
in
Corollary
13.4
necessarily
factors
through
Π
X
K
.
The
proof
of
this
next
step
is
the
main
topic
of
the
following
Section.
Note
that
if
this
fact
(i.e.,
that
θ
factors
through
Π
X
K
)
could
be
proven
more
directly,
then
this
paper
could
be
simplified
con-
siderably.
(For
instance,
Section
11,
as
well
as
the
rather
technical
notions
of
“irreducibly
splittable”
and
“F
I-geometric”
would
be
unnecessary.)
Unfortunately,
however,
the
proof
of
this
fact
in
Section
14
relies
heavily
on
the
“preservation
of
relations”
(Corollary
13.4).
Section
14:
The
Annihilation
of
Inertia
In
this
Section,
we
prove
that
any
continuous
surjective
homomorphism
θ
:
Π
U
K
→
Π
Y
K
over
Γ
K
necessarily
(acts
as
though
it)
factors
through
Π
X
K
.
In
the
process
of
doing
this,
we
complete
the
proof
of
the
(first
part
of)
the
first
main
theorem
of
this
paper,
in
the
case
where
the
base
field
is
a
local
field.
Throughout
most
of
this
Section
(except
for
the
very
end
–
i.e.,
from
the
statement
of
Theorem
14.1
on),
we
continue
to
use
the
notation
of
the
preceding
three
Sections.
Let
us
assume,
moreover,
that
Y
K
is
not
hyperelliptic.
Thus,
it
follows
(e.g.,
from
Lemma
10.4
(3))
that
any
connected
finite
étale
covering
of
Y
K
is
also
non-hyperelliptic.
Let
us
first
consider
the
morphism
→
F
∧
H
0
(Y,
ω
Y
/O
K
)
⊗
O
K
O
∞
K
69
constructed
in
Section
12.
We
would
like
to
show
first
of
all
that
F
F
⊆
F
∧
,
where
F
F
def
⊗
O
K
O
(∗)
F
X
This
morphism
factors
through
F
X
∞
X
=
K
0
H
(X,
ω
X/O
K
)
(cf.
the
discussion
preceding
Proposition
11.3).
To
do
this,
we
argue
as
follows:
Let
x
∈
X
K
(K).
Let
∧
x
:
F
∞
→
O
K
be
the
morphism
induced
by
restriction:
i.e.,
elements
of
F
∞
are
sections
of
L[∞],
hence
differentials
on
X[∞];
thus,
by
taking
the
residue
of
such
a
differential
at
x,
we
get
a
residue
map
F
∞
→
O
K
;
then
taking
the
p-adic
completion
of
this
residue
map
gives
us
x
.
We
would
like
to
show
in
the
following
that
the
restriction
ζ
x
:
H
0
(Y,
ω
Y
/O
K
)
→
O
K
of
x
to
H
0
(Y,
ω
Y
/O
K
)
is
zero.
If
we
show
this,
then
this
will
be
also
hold
for
all
x
∈
F
X
K
(K
)
(where
K
is
a
finite
extension
of
K),
hence
(∗)
F
X
will
follow
immediately
(from
F
consists
precisely
of
all
those
differentials
whose
residues
at
every
point
the
fact
that
F
X
are
zero).
Thus,
let
us
assume
that
ζ
x
=
0.
Let
I
x
⊆
Δ
U
denote
the
inertia
group
(well-defined
up
to
conjugation)
corresponding
to
x.
Then
observe
that
the
restriction
I
x
⊆
Δ
U
→
Δ
Y
of
θ
to
Δ
U
is
nontrivial.
Indeed,
if
this
restriction
were
zero,
then
it
is
clear
from
the
way
that
ζ
x
was
constructed
(cf.
the
“comparison
theorem”
of
Proposition
11.3)
that
ζ
x
would
be
zero.
Thus,
θ(I
x
)
=
{1}.
In
particular,
it
follows
(by
possibly
enlarging
K)
that
there
exists
an
open
normal
subgroup
Π
Y
⊆
Π
Y
K
(corresponding
to
a
covering
Y
K
→
Y
K
)
such
K
that:
(i)
Π
Y
surjects
onto
Γ
K
;
and
(ii)
if
we
let
U
K
→
U
K
be
the
result
of
pulling
back
K
Y
K
→
Y
K
via
θ,
then
U
K
→
U
K
is
a
connected
Galois
covering
which
is
ramified
over
x.
def
Let
G
=
Π
Y
K
/Π
Y
.
Thus,
G
is
a
finite
group,
and
G
=
Gal(U
K
/U
K
)
=
Gal(Y
K
/Y
K
).
K
Note
that
U
K
→
U
K
extends
to
a
ramified
covering
X
K
→
X
K
.
Let
x
∈
X
K
(K)
(where
we
enlarge
K
if
necessary)
be
a
point
lying
above
x.
Now
let
us
denote
with
primes
the
objects
corresponding
to
U
K
,
X
K
,
and
Y
K
that
are
analogous
to
the
objects
already
constructed
for
U
K
,
X
K
,
and
Y
K
.
Thus,
we
have
∧
x
:
(F
∞
)
→
O
K
and
ζ
x
:
H
0
(Y
,
ω
Y
/O
K
)
→
O
K
70
Note
that
the
fact
that
ζ
x
=
0
implies
(since
ζ
x
|
H
0
(Y,ω
Y
/O
K
)
=
ζ
x
)
that
ζ
x
=
0.
Let
σ
∈
G
be
such
that
σ(x
)
=
x
,
σ
=
1.
Then
the
fact
that
σ
fixes
x
implies
that
σ
fixes
x
,
∧
)
is
G-equivariant)
which,
in
turn,
implies
(since
the
inclusion
H
0
(Y
,
ω
Y
/O
K
)
→
(F
∞
that
σ
fixes
ζ
x
.
On
the
other
hand,
by
Corollary
13.4
(“preservation
of
relations”)
and
the
fact
def
that
Y
K
is
not
hyperelliptic,
it
follows
that
the
point
of
the
projective
space
P
K
=
0
P(H
(Y
,
ω
Y
/O
K
))
defined
by
ζ
x
lies
on
the
canonically
embedded
curve
Y
K
⊆
P
K
.
Thus,
the
fact
that
ζ
is
fixed
by
σ
implies
that
Y
admits
a
K-valued
point
which
is
fixed
x
K
by
σ.
Since
Y
K
→
Y
K
=
Y
K
/G
is
étale,
however,
this
is
absurd.
This
contradiction
thus
F
completes
the
proof
that
ζ
x
=
0,
and
hence
also
the
proof
of
(∗)
F
X
.
Let
us
review
what
we
have
done
so
far.
Given
a
surjective
continuous
homomorphism
θ
:
Π
U
K
→
Π
Y
K
over
Γ
K
,
we
have
seen
that
θ
induces
an
injection
H
0
(Y
K
,
ω
Y
K
/K
)
→
H
0
(X
K
,
ω
X
K
/K
)
that
preserves
relations.
Thus,
it
follows
(by
using
the
canonical
embedding
of
Y
K
)
that
we
get
a
dominant
morphism
θ
U
:
U
K
→
Y
K
which
extends
to
a
morphism
θ
X
:
X
K
→
Y
K
by
properness.
Moreover,
given
any
finite
Galois
étale
covering
Y
K
→
Y
K
whose
geometric
part
has
p-power
order,
we
can
pull-back
this
covering
via
θ
to
obtain
U
K
→
U
K
(finite
étale),
X
K
→
X
K
(finite
and
possibly
ramified),
together
with
θ
:
Π
U
→
Π
Y
.
If
K
K
we
then
repeat
the
argument
just
applied
to
θ
for
θ
,
we
see
that
we
get
a
morphism
θ
X
If
we
then
continue
this
procedure
for
arbitrary
:
X
K
→
Y
K
which
lies
over
θ
X
.
finite
Galois
étale
coverings
(whose
geometric
parts
have
p-power
order),
the
well-known
correspondence
between
fundamental
groups
and
categories
of
étale
coverings
thus
shows
that
the
morphism
induced
by
θ
U
on
fundamental
groups
coincides
with
θ
up
to
composition
with
an
inner
automorphism
induced
by
an
element
of
Π
Y
K
.
On
the
other
hand,
since
both
θ
and
the
morphism
induced
by
θ
U
on
fundamental
groups
have
the
property
that
they
lie
over
Γ
K
,
it
follows
that
the
element
of
Π
Y
K
in
question
must
map
to
the
the
center
of
Γ
K
.
Since,
however,
Γ
K
is
center-free
(see
Lemma
15.6
below
–
one
checks
easily
that
there
are
no
“vicious
circles”
in
the
reasoning),
it
follows
that
the
element
in
question
must
therefore
be
an
element
of
Δ
Y
⊆
Π
Y
K
.
That
is
to
say,
we
have
essentially
proven
the
following
result:
Theorem
14.1.
Let
K
be
a
finite
extension
of
Q
p
.
Let
Y
K
be
a
hyperbolic
curve
(not
necessarily
proper)
over
K.
Let
U
K
be
the
spectrum
of
a
one-dimensional
function
field
over
K.
Let
Hom
dom
K
(U
K
,
Y
K
)
be
the
set
of
dominant
K-morphisms
from
U
K
to
Y
K
.
Let
(Π
,
Π
Hom
open
U
K
Y
K
)
be
the
set
of
open,
continuous
group
homomorphisms
Π
U
K
→
Π
Y
K
Γ
K
over
Γ
K
,
considered
up
to
composition
with
an
inner
automorphism
arising
from
Δ
Y
.
Then
the
natural
map
71
open
Hom
dom
K
(U
K
,
Y
K
)
→
Hom
Γ
K
(Π
U
K
,
Π
Y
K
)
is
bijective.
Proof.
We
begin
by
proving
that
this
map
is
injective.
First,
observe
that
by
replacing
U
K
and
Y
K
by
coverings
defined
by
“p-derivates”
(as
in
Definition
0.2)
of
Δ
U
and
Δ
Y
(where
we
use
p-derivates
since
they
are
natural),
we
may
assume
that
both
U
K
and
Y
K
admit
hyperbolic
compactifications
U
K
and
Y
K
(i.e.,
U
K
and
Y
K
are
proper
hyperbolic
curves
over
K).
Since
dominant
K-morphisms
U
K
→
Y
K
are
the
same
as
dominant
K-morphisms
U
K
→
Y
K
,
it
suffices
to
show
that
such
a
morphism
U
K
→
Y
K
is
determined
by
its
induced
morphism
Δ
ab
→
Δ
ab
(on
abelianizations
of
geometric
fundamental
groups).
But
U
Y
this
follows
from
the
fact
that
this
morphism
Δ
ab
→
Δ
ab
clearly
determines
the
morphism
U
Y
between
all
p-power
torsion
points
of
the
Jacobians
of
U
K
and
Y
K
,
hence
it
determines
the
induced
morphism
between
the
Jacobians
of
U
K
and
Y
K
.
Moreover,
since
U
K
and
Y
K
are
both
proper
hyperbolic
(i.e.,
of
genus
≥
2),
they
both
embed
in
their
Jacobians,
so
we
conclude
that
the
original
morphism
U
K
→
Y
K
is
uniquely
determined,
as
desired.
Next,
we
consider
surjectivity.
Let
us
first
consider
the
case
where
Y
K
is
proper.
By
replacing
U
K
by
a
finite
étale
covering
of
U
K
(and
then
descending
at
the
end,
which
is
possible
since
(by
the
preceding
paragraph)
the
natural
map
in
the
Theorem
is
injective),
we
can
assume
that
the
proper
model
X
K
of
U
K
is
hyperbolic.
Note,
moreover,
that
if
θ
:
Π
U
K
→
Π
Y
K
is
open,
then
its
image
is
of
finite
index,
so
by
replacing
Y
K
by
a
finite
étale
covering
of
Y
K
,
it
is
clear
that
we
may
assume
that
θ
is
surjective.
Finally,
by
Lemma
10.4
(4),
by
replacing
Y
K
by
a
finite
étale
covering
of
Y
K
,
it
is
clear
that
we
may
assume
that
Y
K
is
non-hyperelliptic.
Now
we
are
in
the
circumstances
considered
above,
and
so
we
see
that
θ
arises
from
a
geometric
morphism
U
K
→
Y
K
,
as
desired.
This
completes
the
proof
of
the
Theorem
when
Y
K
is
proper.
Now
let
us
consider
the
case
when
Y
K
is
not
proper.
First
note
that
by
replacing
Y
K
by
a
finite
étale
covering
of
Y
K
,
we
may
assume
that
the
compactification
Z
K
of
Y
K
is
hyperbolic.
Now
the
point
is
to
compose
the
given
θ
:
Π
U
K
→
Π
Y
K
with
the
morphism
Π
Y
K
→
Π
Z
K
arising
from
the
compactification
inclusion
Y
K
⊆
Z
K
.
Since
we
know
the
Theorem
to
be
true
for
morphisms
between
U
K
and
Z
K
,
we
thus
obtain
that
Π
U
K
→
Π
Z
K
arises
from
some
dominant
U
K
→
Z
K
(which
necessarily
factors
–
since
U
K
is
the
spectrum
of
a
field
–
through
Y
K
,
thus
yielding
a
U
K
→
Y
K
).
Thus
it
remains
only
to
see
that
the
morphism
induced
on
π
1
’s
by
this
U
K
→
Y
K
is
the
same
as
the
given
θ.
But
this
is
done
(as
usual)
by
considering
a
finite
étale
covering
Y
K
→
Y
K
,
and
applying
the
argument
just
described
to
U
K
and
Y
K
to
obtain
a
U
K
→
Y
K
which
lies
over
the
U
K
→
Y
K
constructed
previously.
As
usual,
this
is
enough
to
show
(cf.
the
argument
directly
preceding
the
statement
of
Theorem
14.1)
that
the
morphism
induced
on
fundamental
groups
by
the
U
K
→
Y
K
that
we
constructed
is
the
same
as
the
given
θ.
This
completes
the
proof
of
the
Theorem.
In
fact,
we
have
the
following:
72
Corollary
14.2.
Let
K
be
a
finite
extension
of
Q
p
.
Let
X
K
(respectively,
Y
K
)
be
either
a
hyperbolic
curve
(not
necessarily
proper!)
over
K
or
the
spectrum
of
a
one-dimensional
function
field
over
K.
Let
Hom
dom
K
(X
K
,
Y
K
)
be
the
set
of
dominant
K-morphisms
from
X
K
to
Y
K
.
Let
Hom
open
(Π
,
Π
X
K
Y
K
)
be
the
set
of
open,
continuous
group
homomorphisms
Γ
K
Π
X
K
→
Π
Y
K
over
Γ
K
,
considered
up
to
composition
with
an
inner
automorphism
arising
from
Δ
Y
.
Then
the
natural
map
open
Hom
dom
K
(X
K
,
Y
K
)
→
Hom
Γ
K
(Π
X
K
,
Π
Y
K
)
is
bijective.
Proof.
By
an
argument
analogous
to
that
of
the
final
paragraph
of
the
proof
of
Theorem
14.1,
it
follows
that
it
suffices
to
consider
the
case
where
Y
K
is
a
hyperbolic
curve.
If
U
K
is
the
generic
point
(considered
as
a
scheme)
of
X
K
,
then
given
any
open
θ
X
:
Π
X
K
→
Π
Y
K
,
composing
θ
X
with
the
natural
surjection
Π
U
K
→
Π
X
K
induced
by
U
K
→
X
K
gives
us
an
open
θ
U
:
Π
U
K
→
Π
Y
K
.
Applying
Theorem
14.1
to
θ
U
thus
gives
us
a
dominant
morphism
X
K
→
Y
K
(where
Y
K
is
the
compactification
of
the
curve
Y
K
).
To
see
that
this
morphism
factors
through
Y
K
,
it
suffices
to
apply
this
construction
to
Y
K
→
Y
K
,
where
Y
K
is
a
finite,
geometrically
connected
(over
K),
Galois
étale
covering
(of
p-power
order)
of
Y
K
which
is
ramified
over
all
the
points
of
Y
K
−
Y
K
.
In
fact,
we
may
even
choose
Y
K
so
that
the
ramification
indices
over
all
the
points
of
Y
K
−
Y
K
are
larger
than
the
degree
of
X
K
over
Y
K
.
Then
the
fact
that
we
get
some
X
K
→
Y
K
(where
X
K
is
étale
over
X
K
)
lying
over
the
morphism
X
K
→
Y
K
obtained
previously
shows
that
this
morphism
X
K
→
Y
K
factors
through
Y
K
,
as
desired.
This
completes
the
proof
of
the
Corollary.
Remark.
Corollary
14.2
is
thus
a
special
case
of
Theorem
A,
the
first
main
theorem
of
this
paper.
In
fact,
Corollary
14.2
holds
even
in
the
case
where
K
is
only
finitely
generated
over
Q
p
.
However,
unlike
the
situation
in
[Mzk2],
deriving
this
finitely
generated
case
from
the
local
field
case
is
not
so
trivial,
again
(cf.
the
discussion
at
the
end
of
Section
13)
because
of
the
fact
that
Δ
U
is
so
large.
Thus,
we
save
the
derivation
of
the
finitely
generated
case
for
the
following
Section.
Section
15:
Base
Fields
Finitely
Generated
over
the
p-adics
In
this
Section,
we
let
L
be
a
finitely
generated
extension
of
Q
p
.
We
would
like
to
prove
versions
of
Theorem
14.1
and
Corollary
14.2
in
the
case
where
the
base
field
is
L
(as
opposed
to
a
finite
extension
of
Q
p
).
Thus,
by
induction
on
the
transcendence
degree
of
L
over
Q
p
,
we
may
assume
that
Theorem
14.1,
for
instance,
is
known
for
all
base
fields
whose
transcendence
degree
over
Q
p
is
<
that
of
L.
Let
X
L
and
Y
L
be
proper
hyperbolic
curves
over
L.
Let
U
L
be
the
generic
point
of
X
L
(regarded
as
a
scheme).
Moreover,
let
us
assume
that
we
have
been
given
a
continuous
surjective
homomorphism
over
Γ
L
73
θ
:
Π
U
L
→
Π
Y
L
Once
it
is
shown
that
θ
factors
though
Π
X
L
,
it
is
relatively
easy
to
derive
that
θ
arises
geometrically
from
Theorem
14.1.
Thus,
the
first
order
of
business
is
to
show
that
θ
factors
through
Π
X
L
.
To
achieve
this,
we
assume
that
this
is
not
the
case.
Then
(cf.
the
argument
employed
in
Section
14)
by
replacing
Y
L
and
U
L
by
finite
étale
coverings
Y
L
→
Y
L
and
U
L
→
U
L
(the
latter
obtained
by
pulling
back
the
former
via
θ),
we
may
assume
that
we
are
in
the
following
situation:
There
is
a
finite
cyclic
group
G
(with
generator
σ)
acting
faithfully
on
U
L
(hence
also
X
L
)
and
Y
L
–
via
L-linear
automorphisms
–
in
such
a
way
that
σ
fixes
a
point
x
0
∈
X
L
(L),
but
acts
without
fixed
points
on
Y
L
.
Moreover,
we
assume
that
σ
◦
θ
◦
σ
−1
coincides
with
θ
up
to
composition
with
an
inner
automorphism
defined
by
an
element
of
Δ
Y
.
If
we
can
show
that
these
assumptions
lead
to
a
contradiction,
it
will
follow
immediately
that
θ
factors
through
Π
X
L
.
Let
(as
usual)
H
U
(respectively,
H
Y
)
be
the
abelianization
of
Δ
U
(respectively,
Δ
Y
)
regarded
as
a
Γ
K
-module.
Let
H
U
P
⊆
H
U
be
the
closure
of
the
image
of
all
the
inertia
groups
(cf.
the
beginning
of
Section
11).
Note
that
θ
induces
a
Γ
K
-morphism
H
U
→
H
Y
.
Since
H
Y
is
a
finitely
generated
Z
p
-module,
it
follows
that
(by
possibly
replacing
L
by
a
finite
extension
of
L)
there
exist
points
x
1
,
.
.
.
,
x
r
∈
X
L
(L)
such
that
the
images
of
the
corresponding
inertia
groups
θ(I
i
)
(for
i
=
1,
.
.
.
,
r)
in
H
Y
is
equal
to
the
image
of
θ(H
U
P
)
in
H
Y
.
Now
let
K
⊆
L
be
a
subfield
such
that
L
is
a
one-dimensional
function
field
over
K
(hence,
in
particular,
we
assume
that
K
is
algebraically
closed
in
L).
Note
that
such
a
K
always
exists
(as
long
as
L
is
not
a
finite
extension
of
Q
p
).
Thus,
there
exists
a
smooth
affine
model
M
→
Spec(K)
of
L
such
that
X
L
and
Y
L
extend
to
smooth
curves
X
M
→
M
and
Y
M
→
M
over
M.
Moreover,
we
may
also
assume
(by
shrinking
M)
that
x
1
,
.
.
.
,
x
r
extend
to
sections
s
1
,
.
.
.
,
s
r
:
X
M
→
M
whose
images
are
disjoint
from
one
another,
and
that
σ
acts
on
X
M
and
Y
M
.
The
point
x
0
∈
X
L
(L)
then
extends
(by
the
valuative
criterion
for
properness)
to
a
section
s
0
:
M
→
X
M
.
Finally,
let
us
observe
that
(by
further
shrinking
M),
we
may
assume
that
σ
acts
without
fixed
points
on
Y
M
.
The
first
thing
that
we
would
like
to
get
our
hands
on
is
a
morphism
like
θ,
except
for
objects
over
K,
so
that
we
can
apply
Theorem
14.1
over
K
(which
we
know
to
be
true
via
the
induction
hypothesis).
To
construct
such
a
morphism,
we
argue
as
follows.
Let
m
∈
M(K)
be
a
point
(which,
after
possibly
enlarging
K,
always
exists).
Let
D
be
the
spectrum
of
the
completion
of
O
M,m
.
(Here,
one
should
think
of
the
“D”
as
standing
for
“disk.”)
Thus,
we
may
choose
an
isomorphism
D
∼
=
Spec(K[[t]])
(where
t
is
an
indeterminate).
Let
D
∞
→
D
1
be
the
ramified
covering
of
infinite
degree
obtained
by
adjoining
all
t
N
(for
N
a
positive
integer)
to
O
D
.
Let
p
∈
X
M
be
the
prime
which
is
the
fiber
X
m
of
X
M
→
M
over
M.
Let
U
D
be
the
spectrum
of
the
completion
of
O
X
M
,
p
.
Thus,
U
D
is
the
spectrum
of
a
complete
discrete
valuation
ring
with
residue
field
equal
to
K(U
m
),
the
function
field
of
X
m
.
Let
η
D
def
(respectively,
η
D
∞
)
be
the
generic
point
of
D
(respectively,
D
∞
).
Let
U
η
D
=
U
D
×
D
η
D
.
Thus,
U
η
D
is
the
spectrum
of
a
complete,
discretely
valued
field.
In
particular,
it
follows
74
from
the
well-known
theory
of
such
fields
(see,
e.g.,
[Ser2])
that
if
Γ
U
ηD
is
the
absolute
Galois
group
of
this
field,
then
we
have
an
exact
sequence
1
→
Z(1)
→
Γ
U
ηD
→
Γ
K(U
m
)
→
1
Here
the
covering
corresponding
to
the
Z(1)
is
given
by
adjoining
all
t
N
(for
N
a
positive
integer)
–
cf.
the
covering
D
∞
→
D.
1
Now
let
us
denote
by
Π
U
η
the
result
of
replacing
the
geometric
portion
(i.e.,
relative
D
to
the
morphism
U
η
D
→
η
D
)
of
the
fundamental
group
of
U
η
D
by
its
maximal
pro-p
quotient.
Then
we
get
an
exact
sequence
1
→
Δ
U
m
→
Π
U
η
D
→
Γ
η
D
→
1
If
we
pull
this
exact
sequence
back
via
Γ
η
D
∞
=
Γ
K
→
Γ
η
D
,
we
thus
get
an
exact
sequence
1
→
Δ
U
m
→
Π
U
η
D
∞
→
Γ
η
D
∞
=
Γ
K
→
1
Moreover,
it
is
easy
to
see
that
this
last
exact
sequence
can
be
identified
with
1
→
Δ
U
m
→
Π
U
m
→
Γ
K
→
1
On
the
other
hand,
if
we
pull
back
Π
Y
L
→
Γ
L
via
Γ
K
=
Γ
η
D
∞
→
Γ
L
,
we
get
an
exact
sequence
1
→
Δ
Y
m
→
Π
Y
m
→
Γ
K
→
1
Thus,
if
we
pull-back
θ
via
Γ
K
=
Γ
η
D
∞
→
Γ
L
to
obtain
a
morphism
Π
U
η
D
∞
→
Π
Y
η
D
∞
(where
the
subscripted
η
D
∞
denotes
“⊗
L
η
D
∞
”)
and
compose
with
“π
1
”
of
the
natural
morphism
U
η
D
∞
→
U
η
D
∞
,
we
get
a
continuous
homomorphism
θ
m
:
Π
U
m
→
Π
Y
m
over
Γ
K
(where
K
we
regard
here
as
the
residue
field
of
M
at
m).
Lemma
15.1.
The
morphism
θ
m
is
surjective.
Proof.
It
suffices
to
show
that
the
restriction
of
θ
m
to
Δ
U
m
surjects
onto
Δ
Y
m
=
Δ
Y
.
In
fact,
by
the
basic
theory
of
p-groups,
it
suffices
to
show
that
θ
m
induces
a
surjection
of
Δ
U
m
onto
H
Y
m
=
H
Y
.
To
see
this,
it
suffices
to
consider
(after
replacing
M
by
a
finite
75
étale
covering
of
M,
and
enlarging
K
if
necessary)
a
finite
abelian
covering
Y
L
→
Y
L
(where
Y
L
→
L
is
geometrically
connected)
of
degree
a
power
of
p
which
is
>
1.
Let
us
pull-back
this
covering
to
some
covering
U
L
→
U
L
via
θ.
Then
we
must
show
that
the
pull-back
U
η
D
∞
→
U
η
D
∞
of
this
covering
to
U
η
D
∞
is
not
the
trivial
covering.
Thus,
suppose
that
it
is
the
trivial
covering.
Then
the
covering
U
m
→
U
m
that
it
induces
(cf.
the
exact
sequences
above)
of
U
m
is
again
trivial.
Since
the
images
of
s
1
,
.
.
.
,
s
r
in
X
m
are
disjoint
this
implies
first
of
all
that
U
L
→
U
L
is
unramified
at
x
1
,
.
.
.
,
x
r
.
But
because
of
the
way
in
which
x
1
,
.
.
.
,
x
r
were
chosen,
this
implies
that
the
covering
Y
L
→
Y
L
was
obtained
from
a
quotient
of
H
Y
/Im(θ(H
U
P
)).
Thus,
it
follows
that
U
L
→
U
L
extends
to
a
finite
étale
covering
X
L
→
X
L
.
Moreover,
since
X
M
is
smooth
over
M
at
m,
it
follows
that
X
L
→
X
L
is
split
if
and
only
if
the
induced
X
m
→
X
m
is
split.
But
this
X
m
→
X
m
extends
the
covering
U
m
→
U
m
which
we
already
saw
to
be
trivial.
Thus,
we
obtain
that
X
L
→
X
L
,
hence
U
L
→
U
L
is
trivial.
Since
θ
is
surjective,
however,
this
implies
that
Y
L
→
Y
L
is
trivial.
This
contradiction
completes
the
proof
of
the
Lemma.
Now
by
the
induction
hypothesis
on
the
transcendence
degree
of
L,
it
follows
that
θ
m
arises
from
some
geometric
morphism
U
m
→
Y
m
which
is
compatible
with
the
action
of
σ
on
both
sides
(since
σ
is
compatible
with
θ).
By
the
valuative
criterion
for
properness,
this
morphism
extends
to
a
morphism
X
m
→
Y
m
.
Moreover,
σ
fixes
s
0
(m)
∈
X
m
(K),
so
s
0
(m)
∈
X
m
(K)
is
mapped
to
a
fixed
point
of
Y
m
(K),
which
is
absurd,
since
σ
acts
on
Y
m
without
fixed
points.
This
contradiction
completes
the
proof
of
the
following
result:
Lemma
15.2.
Let
L
be
a
finitely
generated
extension
of
Q
p
.
Let
X
L
and
Y
L
be
proper
hyperbolic
curves
over
L.
Let
U
L
be
the
generic
point
of
X
L
.
Then
any
continuous
surjective
homomorphism
θ
:
Π
U
L
→
Π
Y
L
over
Γ
L
necessarily
factors
through
Π
X
L
.
Now
we
can
conclude
that
θ
arises
geometrically,
as
follows.
Consider
the
M-scheme
def
H
M
=
Hom
M
(X
M
,
Y
M
)
→
M.
Since
Y
M
is
hyperbolic,
it
is
well-known
that
H
M
is
finite
and
unramified
over
M.
(Indeed,
that
H
M
→
M
is
unramified
(respectively,
quasi-
finite;
proper)
follows
since
the
pull-back
of
the
tangent
bundle
of
Y
m
to
X
m
(for
any
m
∈
M)
has
no
global
sections
(respectively,
follows
from
the
Hurwitz
formula,
which
allows
one
to
bound
the
degree
of
a
morphism
X
m
→
Y
m
;
follows
via
the
same
argument
as
that
used
in
Lemma
8.3
of
[Mzk1]).)
By
shrinking
M,
we
may
assume
that
H
M
is
finite
étale
over
M.
Then
the
fact
that
θ
m
arises
geometrically
(from
some
X
m
→
Y
m
)
shows
that
over
some
finite
extension
L
of
L,
we
have
a
morphism
X
L
→
Y
L
that
specializes
to
X
m
→
Y
m
.
Moreover,
if
M
is
the
normalization
of
M
in
L
,
then
since
the
pro-p
geometric
fundamental
groups
of
X
M
and
Y
M
form
local
systems
over
M
,
it
follows
(by
checking
what
happens
over
the
point
m)
that
the
morphism
on
Δ’s
induced
by
X
L
→
Y
L
is
the
same
(up
to
composition
with
an
inner
automorphism)
as
that
induced
by
θ.
In
particular,
the
morphism
H
X
→
H
Y
induced
by
X
L
→
Y
L
is
the
same
as
that
induced
by
θ.
On
the
other
hand,
a
morphism
from
X
L
to
Y
L
(over
any
field)
is
determined
by
the
morphism
it
induces
from
H
X
to
H
Y
(cf.
the
proof
of
the
injectivity
part
of
Theorem
14.1).
Thus,
since
the
H
X
→
H
Y
in
question
is
Γ
L
-
(not
just
Γ
L
-)
equivariant,
we
obtain
76
that
X
L
→
Y
L
descends
to
a
X
L
→
Y
L
.
By
repeating
this
construction
(as
usual)
for
all
finite
étale
coverings
of
Y
L
,
we
also
obtain
that
the
morphism
induced
on
fundamental
groups
by
this
X
L
→
Y
L
is
the
original
θ.
That
is
to
say,
in
summary,
we
have
proven
that
any
θ
:
Π
U
L
→
Π
Y
L
as
in
Lemma
15.2
necessarily
arises
geometrically
from
some
morphism
U
L
→
Y
L
.
Thus,
by
arguments
formally
analogous
to
those
of
the
proofs
of
Theorem
14.1
and
Corollary
14.2,
we
obtain
the
following
analogue
of
Corollary
14.2:
Corollary
15.3.
Let
K
be
a
finitely
generated
field
extension
of
Q
p
.
Let
X
K
(re-
spectively,
Y
K
)
be
either
a
hyperbolic
curve
over
K
or
the
spectrum
of
a
one-dimensional
function
field
over
K.
Let
Hom
dom
K
(X
K
,
Y
K
)
be
the
set
of
dominant
K-morphisms
from
X
K
to
Y
K
.
Let
Hom
open
(Π
,
Π
X
K
Y
K
)
be
the
set
of
open,
continuous
group
homomorphisms
Γ
K
Π
X
K
→
Π
Y
K
over
Γ
K
,
considered
up
to
composition
with
an
inner
automorphism
arising
from
Δ
Y
.
Then
the
natural
map
open
Hom
dom
K
(X
K
,
Y
K
)
→
Hom
Γ
K
(Π
X
K
,
Π
Y
K
)
is
bijective.
It
turns
out
that
it
is
most
natural
to
generalize
Corollary
15.3
to
the
case
where
the
long
and
unwieldy
expression
“hyperbolic
curve/spectrum
of
a
one-dimensional
function
field
defined
over
a
finitely
generated
field
extension
of
Q
p
”
is
replaced
by
the
much
shorter
expression
“pro-hyperbolic
curve
over
a
sub-p-adic
field.”
Thus,
we
make
the
following
definition:
Let
K
be
a
field.
Let
X
K
be
a
K-scheme.
Definition
15.4.
(i)
We
shall
call
a
field
K
a
sub-p-adic
field
if
there
exists
a
prime
num-
ber
p,
together
with
a
finitely
generated
field
extension
L
of
Q
p
such
that
K
is
isomorphic
to
a
subfield
of
L.
(ii)
We
shall
call
X
K
a
hyperbolic
pro-curve
(over
K)
if
X
K
can
be
written
as
the
projec-
tive
limit
of
a
projective
system
of
hyperbolic
curves
over
K
such
that
all
the
transition
morphisms
in
the
projective
system
are
birational.
Thus,
the
following
are
all
examples
of
sub-p-adic
fields:
(1)
finitely
generated
(in
particular,
finite)
extensions
of
Q
p
(2)
number
fields
(i.e.,
finite
extensions
of
Q)
(3)
the
subfield
of
Q
which
is
the
composite
of
all
number
fields
of
degree
≤
n
over
Q
(for
some
fixed
integer
n).
77
Another
way
to
think
of
a
pro-hyperbolic
curve
X
K
is
as
the
result
of
removing
some
set
(possibly
infinite,
possibly
empty)
of
closed
points
from
some
hyperbolic
curve.
In
partic-
ular,
the
notion
of
a
“hyperbolic
pro-curve”
generalizes
both
the
notion
of
a
hyperbolic
curve
and
the
case
of
the
spectrum
of
a
function
field
of
dimension
one.
Then
we
have
the
following
result:
Corollary
15.5.
Let
K
be
sub-p-adic.
Let
X
K
(respectively,
Y
K
)
be
a
hyperbolic
pro-curve
over
K.
Let
Hom
dom
K
(X
K
,
Y
K
)
be
the
set
of
dominant
K-morphisms
from
X
K
open
to
Y
K
.
Let
Hom
Γ
K
(Π
X
K
,
Π
Y
K
)
be
the
set
of
open,
continuous
group
homomorphisms
Π
X
K
→
Π
Y
K
over
Γ
K
,
considered
up
to
composition
with
an
inner
automorphism
arising
from
Δ
Y
.
Then
the
natural
map
open
Hom
dom
K
(X
K
,
Y
K
)
→
Hom
Γ
K
(Π
X
K
,
Π
Y
K
)
is
bijective.
Proof.
First,
let
us
observe
that
if
K
is
a
finitely
generated
field
extension
of
Q
p
,
then
Corollary
15.5
follows
immediately
from
Corollary
15.3
by
arguments
formally
analogous
to
those
of
the
proofs
of
Theorem
14.1
and
Corollary
14.2.
Thus,
the
only
(slightly)
“new”
phenomenon
here
is
the
fact
that
we
allow
K
to
be
a
subfield
of
a
finitely
generated
extension
of
Q
p
.
The
argument
for
such
subfields
is
as
follows:
Let
K
be
a
subfield
of
a
finitely
generated
extension
field
L
of
Q
p
.
Suppose
that
we
have
been
given
X
K
and
Y
K
as
in
the
statement
of
Corollary
15.5,
as
well
as
an
open
θ
:
Π
X
K
→
Π
Y
K
over
Γ
K
.
Let
us
denote
by
θ
geom
the
morphism
Δ
X
→
Δ
Y
induced
by
θ
on
geometric
fundamental
groups.
By
base-changing
to
L
and
applying
the
Corollary
15.5
over
L,
we
obtain
that
there
exists
a
morphism
φ
L
:
X
L
→
Y
L
whose
induced
morphism
on
geometric
fundamental
groups
coincides
with
that
defined
by
θ.
On
the
other
hand,
since
morphisms
between
hyperbolic
curves
(hence
also
hyperbolic
pro-curves),
clearly
have
no
moduli
(cf.
the
fact
that
the
scheme
H
M
of
the
discussion
preceding
Corollary
15.3
was
finite
and
unramified
over
M),
it
follows
that
φ
L
descends
to
a
finite
Galois
extension
K
of
K.
Thus,
we
have
a
morphism
φ
K
:
X
K
→
Y
K
.
It
remains
to
descend
φ
K
to
K.
But
this
follows
from
the
fact
that
if
σ
∈
Gal(K
/K),
then
conjugating
φ
K
by
σ
gives
a
morphism
φ
σK
:
X
K
→
Y
K
whose
induced
morphism
on
geometric
fundamental
groups
is
the
result
of
conjugating
θ
geom
by
σ.
On
the
other
hand,
θ
geom
arises
from
θ
which
lies
over
Γ
K
,
so
θ
geom
is
fixed
(up
to
composition
with
an
inner
automorphism
defined
by
an
element
of
Δ
Y
)
by
conjugation
by
σ.
Thus,
φ
K
and
φ
σK
induce
the
same
morphism
on
geometric
fundamental
groups,
hence
coincide
(cf.
the
argument
of
the
discussion
preceding
Corollary
15.3).
This
shows
that
φ
K
descends
to
a
morphism
φ
K
:
X
K
→
Y
K
.
Repeating
this
construction
(as
usual
–
cf.
the
argument
preceding
the
statement
of
Theorem
14.1)
for
coverings
of
X
K
and
Y
K
shows
that
the
morphism
induced
by
φ
K
on
fundamental
groups
coincides
with
θ
(up
to
composition
with
an
inner
automorphism
defined
by
an
element
of
Δ
Y
).
Remark.
Note
that
in
Corollaries
15.3
and
15.5,
in
fact,
we
implicitly
used
the
fact
that
78
for
K
as
in
those
Corollaries,
Γ
K
is
center-free.
This
may
be
proven
as
follows.
First,
if
K
is
a
finite
extension
of
Q
p
,
then
we
have
the
following:
Lemma
15.6.
If
K
is
a
finite
extension
of
Q
p
,
then
the
group
Γ
K
is
center-free.
Proof.
(The
argument
given
here
is
“well-known,”
but
was
related
to
the
author
by
A.
Tamagawa.)
Since
one
knows
explicitly
(see,
e.g.,
[Ser2],
Chapter
IV,
§2)
the
structure
of
Gal(K
tm
/K),
it
is
clear
that
the
quotient
Gal(K
tm
/K)
of
Γ
K
is
center-free.
Thus,
the
center
of
Γ
K
must
lie
in
Γ
K
tm
⊆
Γ
K
.
On
the
other
hand,
Γ
K
tm
is
a
pro-p
group.
Moreover,
it
follows
immediately
from
the
facts
that
(1)
H
2
(K,
F
p
(1))
=
F
p
;
(2)
if
K
is
a
finite
unramified
extension
of
K
of
degree
p
(which
always
exists),
then
the
natural
morphism
H
2
(K,
F
p
(1))
→
H
2
(K
,
F
p
(1))
is
zero;
that
H
2
(Γ
K
tm
,
F
p
)
=
0.
But
by
[Shatz],
Chapter
III,
§3,
Proposition
2.3,
this
is
enough
to
imply
that
Γ
K
tm
is
free
(as
a
pro-p
group),
which
implies
that
its
center
is
trivial
(see,
e.g.,
[Tama],
§1,
Propositions
1.1,
1.11).
Next,
we
consider
the
case
of
a
finitely
generated
extension
of
Q
p
:
Lemma
15.7.
Suppose
that
K
0
is
a
field
of
characteristic
zero
with
the
property
that
every
open
subgroup
of
Γ
K
0
is
center-free.
Then
any
finitely
generated
field
extension
K
of
K
0
also
has
this
property.
In
particular,
if
K
is
a
finitely
generated
field
extension
of
Q
p
,
then
Γ
K
is
center-free.
Proof.
The
last
statement
follows
from
the
first
plus
Lemma
15.6.
Thus,
let
us
prove
the
first
statement.
Note
that
an
extension
of
a
center-free
group
by
a
center-free
group
is
center-free.
Thus,
it
suffices
to
prove
that
if
L
is
a
function
field
(of
arbitrary
finite
dimension)
over
K
0
,
then
Γ
L
is
center-free.
Next,
note
that
the
projective
limit
of
a
projective
system
of
center-free
groups
in
which
all
the
transition
morphisms
are
surjective
is
center-free.
Moreover,
as
is
well-known,
Spec(L)
is
a
projective
limit
of
“hyperbolic
Artin
neighborhoods”
over
K
0
(i.e.,
successive
fibrations
of
hyperbolic
curves
–
see
[SGA4],
XI
3.3).
Thus,
it
suffices
to
prove
that
the
fundamental
group
of
such
an
Artin
neighborhood
is
center-free.
But
this
then
reduces
to
showing
that
the
fundamental
group
of
a
hyperbolic
curve
is
center-free,
which
is
well-known
(see,
e.g.,
[Tama],
§1,
Proposition
1.11).
This
is
already
enough
for
Corollary
15.3.
Now
we
can
conclude
the
result
for
arbitrary
K
as
in
Corollary
15.5
by
means
of
the
following
Lemma
(due
to
A.
Tamagawa):
79
Lemma
15.8.
Let
K
be
sub-p-adic.
Then
Γ
K
is
center-free.
Proof.
Let
K
be
a
finitely
generated
field
extension
of
Q
p
that
contains
K.
Suppose
that
σ
∈
Γ
K
lies
in
the
center
of
Γ
K
,
but
is
not
equal
to
the
identity.
Then
there
exists
a
finite
Galois
extension
L
of
K
such
that
σ
maps
to
an
element
of
Gal(L/K)
other
than
the
identity.
Write
L
=
K(α),
for
some
α
∈
L.
Let
E
L
be
an
elliptic
curve
over
L
with
j-invariant
α.
Let
X
L
be
the
complement
of
“0”
in
E
L
.
Thus,
X
L
is
a
hyperbolic
curve
over
L.
Let
Y
L
be
the
result
of
base-changing
X
L
by
σ
:
L
→
L.
Thus,
it
follows
that
X
L
and
Y
L
are
isomorphic
as
schemes
over
K.
Moreover,
conjugating
by
σ
defines
an
isomorphism
Π
X
L
∼
=
Π
Y
L
which
lies
over
Γ
L
because
σ
is
in
the
center
of
Γ
K
.
Base-
changing
this
isomorphism
to
L
(the
composite
of
K
and
L
over
K),
we
get
a
Γ
L
-
isomorphism
Π
X
L
∼
=
Π
Y
L
.
By
Corollary
15.3
(in
fact,
really,
this
follows
already
from
the
results
of
[Mzk2]),
we
obtain
that
this
isomorphism
arises
from
an
L
-isomorphism
of
X
L
with
Y
L
.
But
this
implies
that
the
j-invariants
of
the
compactifying
elliptic
curves
of
X
L
and
Y
L
are
the
same,
i.e.,
that
α
=
α
σ
∈
L
⊆
L
,
which
is
absurd.
This
contradiction
completes
the
proof
of
the
Lemma.
Section
16:
Maps
From
Higher-Dimensional
Function
Fields
to
Curves
Let
K
be
sub-p-adic
(cf.
Definition
15.4
(i)).
Let
U
K
be
the
spectrum
of
a
function
field
over
K.
(Note
that
here,
we
do
not
assume
that
the
dimension
of
the
function
field
is
one.)
Let
X
K
be
a
smooth
projective
model
of
U
K
(which
exists
by
[Hiro]).
Thus,
U
K
is
the
generic
point
of
X
K
.
Let
n
be
the
dimension
of
X
K
.
By
abuse
of
terminology,
we
shall
also
say
that
n
is
the
“dimension
of
U
K
over
K.”
Since
the
one-dimensional
case
has
been
dealt
with
previously,
we
assume
here
that
n
≥
2.
The
purpose
of
this
Section
is
to
prove
a
result
like
Corollary
15.5,
except
for
morphisms
between
U
K
and
a
hyperbolic
pro-curve.
Let
L
be
a
very
ample
line
bundle
on
X
K
.
Let
def
V
=
Γ(X
K
,
L)
Since
the
dimension
of
X
K
is
≥
2,
and
L
is
very
ample,
it
follows
that
dim
K
(V
)
≥
3.
Let
W
⊆
V
be
a
one-dimensional
(over
K)
subspace,
generated
by
a
section
of
L
whose
zero
locus
forms
a
smooth,
connected
subvariety
of
X
K
.
(Note
that
by
Bertini’s
theorem
(see,
e.g.,
Theorem
6.3,
pp.
66,
of
[Jou]),
it
is
well-known
that
such
a
W
exists.)
Let
W
⊆
V
be
a
two-dimensional
subspace
containing
W
.
Choose
a
basis
{e
1
,
e
2
}
for
W
such
that
e
1
∈
W
.
Let
e
3
∈
V
be
an
element
whose
image
in
V
/W
is
nonzero.
Let
us
also
assume
that
the
common
zero
locus
in
X
K
of
the
three
sections
e
1
,
e
2
,
e
3
is
a
subscheme
of
X
K
of
codimension
≥
3.
(This
can
always
be
achieved
by
choosing
e
1
,
e
2
,
e
3
sufficiently
def
def
generically.)
Let
K
=
K(t)
(where
t
is
an
indeterminate).
Let
s
1
=
t;
s
2
=
t
−1
.
Then
we
def
can
construct
a
new
two-dimensional
subspace
W
⊆
V
K
=
V
⊗
K
K
,
over
K
,
as
follows:
80
def
def
We
let
W
be
the
subspace
generated
by
the
vectors
e
1
=
e
1
+
s
1
·
e
3
,
e
2
=
e
2
+
s
2
·
e
3
.
Thus,
to
summarize,
we
have
the
following
situation:
W
⊆
⊆
W
K
·
e
1
+
K
·
e
2
+
K
·
e
3
K
·
e
1
⊆
V
K
·
e
1
+
K
·
e
2
⊆
K
·
e
1
+
K
·
e
2
+
K
·
e
3
e
2
⊆
V
K
=
V
⊗
K
K
;
W
=
K
·
e
1
+
K
·
e
1
=
e
1
+
s
1
·
e
3
;
⊆
def
e
2
=
e
2
+
s
2
·
e
3
;
K
=
K(t)
s
1
=
t;
s
2
=
t
−1
Lemma
16.1.
Let
Ω
be
an
algebraically
closed
field
containing
K
.
Then
there
do
not
def
exist
any
one-dimensional
subspaces
W
of
W
Ω
=
W
⊗
K
Ω
such
that
W
is
defined
(as
a
subspace
of
V
⊗
K
Ω)
over
a
finite
extension
of
K.
Proof.
Indeed,
if
such
a
W
existed,
then
there
would
exist
elements
a,
b,
c
∈
Ω
(with
c
=
0,
and
at
least
one
of
a,
b
nonzero),
together
with
α
1
,
α
2
,
α
3
∈
K
such
that
a
·
e
1
+
b
·
e
2
=
c
·
(α
1
·
e
1
+
α
2
·
e
2
+
α
3
·
e
3
)
By
dividing
a
and
b
by
c,
we
may
assume
that
c
=
1.
Then,
equating
the
coefficients
of
e
1
,
e
2
,
e
3
,
we
obtain
that
a
=
α
1
;
b
=
α
2
;
s
1
·
a
+
s
2
·
b
=
α
3
.
But
this
implies
that
t,
t
−1
,
and
1
(as
elements
of
K
)
are
linearly
dependent
over
K,
which
is
absurd.
This
completes
the
proof
of
the
Lemma.
Before
proceeding,
let
us
pause
to
interpret
what
this
Lemma
means.
Note
that
the
3-dimensional
K-subspace
of
V
generated
by
e
1
,
e
2
,
e
3
defines
a
rational
morphism
ψ
from
def
X
K
to
Q
K
=
P(K
·
e
1
+
K
·
e
2
+
K
·
e
3
).
Thus,
ψ
is
defined
outside
of
some
closed
subscheme
of
X
K
of
codimension
≥
2.
Moreover,
note
that
the
inverse
image
via
ψ
of
any
closed
point
of
Q
K
is
a
subscheme
of
X
K
of
codimension
≥
2.
(Indeed,
this
follows
from
the
fact
that
the
common
zero
locus
of
e
1
,
e
2
,
e
3
has
codimension
≥
3
in
X
K
.)
Now
let
Ω
be
a
finite
extension
of
K
,
and
let
us
consider
a
one-dimensional
(over
Ω)
subspace
W
def
of
W
Ω
.
Thus,
W
corresponds
to
a
line
L
Ω
⊆
Q
Ω
=
Q
K
⊗
K
Ω,
and
the
zero
locus
(in
−1
X
Ω
=
X
K
⊗
K
Ω)
of
a
nonzero
section
of
W
is
equal
to
the
closure
of
ψ
Ω
(L
Ω
)
⊆
X
Ω
.
def
Now
we
are
ready
to
interpret
Lemma
16.1:
Namely,
I
claim
that
Lemma
16.1
implies
−1
(L
Ω
)
is
defined
over
K.
that
no
irreducible
component
of
the
(closure
of
the)
divisor
ψ
Ω
81
Indeed,
if
there
were
a
divisor
D
⊆
X
K
such
that
ψ
Ω
(D
Ω
)
⊆
L
Ω
,
then
by
spreading
out
Ω
(respectively,
L
Ω
)
to
a
smooth
affine
curve
C
K
over
K
(respectively,
a
family
of
lines
in
Q
K
parametrized
by
C
K
),
we
obtain
that
for
every
closed
point
c
∈
C
K
,
we
have
ψ(D)
⊆
L
c
.
But
since
(by
Lemma
16.1)
L
Ω
is
not
defined
over
K,
it
follows
that
for
two
sufficiently
generic
closed
points
c,
c
∈
C
K
,
dim
K
(L
c
L
c
)
=
0,
which
implies
that
D
is
contained
in
the
inverse
image
of
a
finite
set
of
closed
points
of
Q
K
.
But
we
saw
in
the
preceding
paragraph
that
the
inverse
image
via
ψ
of
a
closed
point
of
Q
K
is
of
codimension
≥
2.
Since
D
is
a
divisor,
this
is
absurd.
This
completes
the
proof
of
the
claim
stated
at
the
beginning
of
this
paragraph.
Next,
let
us
base-change
U
K
and
X
K
to
K
:
this
gives
rise
to
U
K
and
X
K
.
The
def
two-dimensional
subspace
W
⊆
V
K
defines
a
rational
map
from
X
K
to
P
K
=
P(
W
).
Let
X
K
⊆
X
K
be
the
complement
of
the
indeterminacy
locus
of
this
rational
map.
Thus,
the
complement
of
X
K
in
X
K
is
of
codimension
≥
2,
and,
moreover,
we
obtain
a
dominant
morphism
(a
“pencil”)
ξ
:
X
K
→
P
K
Let
η
P
be
the
generic
point
of
P
K
.
Let
η
P
be
the
spectrum
of
an
algebraic
closure
of
K(η
P
)
(the
function
field
of
P
K
).
Let
def
F
η
P
⊆
X
η
P
=
X
K
×
K
η
P
be
the
divisor
which
is
the
zero
locus
of
the
section
of
L
⊗
K
K(η
P
)
defined
by
the
generic
point
η
P
of
P
K
.
(Thus,
F
η
P
→
η
P
is
proper.)
Put
another
way,
F
η
P
is
the
(closure
in
X
η
P
of
the)
fiber
(⊆
X
η
P
⊆
X
η
P
)
of
the
morphism
ξ
over
the
generic
point
η
P
∈
P
K
(whence
the
use
of
the
letter
“F
”).
Since
the
pencil
P
K
contains
e
1
,
which
is
a
genericization
of
e
1
(i.e.,
e
1
is
the
specialization
of
e
1
at
s
1
=
t
=
0),
and
the
zero
locus
of
e
1
is
(geometrically)
smooth
and
connected,
it
follows
that
F
η
P
will
also
be
geometrically
smooth
and
connected
over
η
P
.
Moreover,
by
the
Lefshetz
hyperplane
theorem
(see,
e.g.,
[SGA2]),
it
follows
(since
dim
K
(X
K
)
≥
2)
that
the
natural
morphism
π
1
(F
η
P
)
→
π
1
(X
η
P
)
=
π
1
(X
K
)
=
π
1
(X
K
)
is
surjective.
Let
def
G
η
P
=
F
η
P
×
X
ηP
U
η
P
=
F
η
P
×
X
K
U
K
Since
U
K
is
a
projective
limit
of
dense
open
subschemes
of
X
K
,
it
thus
follows
that
G
η
P
is
a
projective
limit
of
open
subschemes
of
F
η
P
.
Moreover,
since
F
η
P
(thought
of
as
(the
closure
of)
the
fiber
of
the
dominant
morphism
ξ
over
the
generic
point
η
P
of
P
K
)
contains
(as
a
dense
open
subset)
a
scheme
which
is
a
projective
limit
of
dense
open
subschemes
of
X
K
,
we
conclude
that
G
η
P
is
an
integral,
nonempty
scheme.
82
At
any
rate,
we
get
a
natural
morphism
π
1
(G
η
P
)
→
π
1
(U
η
P
)
=
π
1
(U
K
)
=
π
1
(U
K
)
Now
we
have
the
following
important
Lemma
16.2.
This
morphism
π
1
(G
η
P
)
→
π
1
(U
K
)
is
surjective.
→
U
(of
degree
>
1),
Proof.
It
suffices
to
take
a
finite,
connected
étale
covering
U
K
K
pull
it
back
to
a
covering
G
η
P
→
G
η
P
over
G
η
P
,
and
show
that
this
pulled
back
covering
η
→
G
η
is
split.
Now
observe
that
since
ξ
is
can
never
be
split.
Indeed,
suppose
that
G
P
P
dominant,
it
follows
that
F
η
P
→
X
K
and
G
η
P
→
U
K
are
dominant.
In
fact,
F
η
P
→
X
K
and
G
η
P
→
U
K
are
even
birational
isomorphisms.
Moreover,
I
claim
that
the
divisors
(in
X
K
)
that
were
thrown
out
of
X
K
to
create
F
η
P
–
let
us
call
these
divisors
F
-divisors
–
are
different
from
the
divisors
that
were
thrown
out
of
X
K
to
create
U
K
–
which
we
shall
call
U-divisors.
Indeed,
as
divisors
of
X
K
=
X
K
⊗
K
K
,
the
U-divisors
are
all
defined
over
finite
extensions
of
K,
whereas
the
F
-divisors
(which
are
just
fibers
of
ξ
over
closed
points
of
P
K
)
are,
by
Lemma
16.1
(see
also
the
interpretation
of
Lemma
16.1
in
the
two
paragraphs
following
the
proof
of
Lemma
16.1),
never
defined
over
finite
extensions
of
K.
This
proves
the
claim.
Next,
let
us
observe
that
the
morphism
F
η
P
→
F
η
P
is
ramified
only
over
F
-divisors.
Thus,
it
is
unramified
over
U-divisors.
Since
G
η
P
→
F
η
P
is
birational,
the
fact
that
η
→
G
η
splits
(hence,
in
particular,
extends
to
a
finite
étale
covering
over
F
η
)
thus
G
P
P
P
→
U
arises
→
U
is
unramified
over
the
U-divisors,
i.e.,
U
implies
that
the
original
U
K
K
K
K
→
X
.
(Here
we
use
“purity
of
the
branch
locus”
for
regular
local
from
a
covering
X
K
K
rings
–
see,
e.g.,
[SGA2],
Exposé
X,
p.
118,
Théorème
3.4.)
But
then
the
fact
that
the
pull-back
F
η
P
→
F
η
P
of
this
covering
to
F
η
P
does
not
split
follows
from
the
surjectivity
observed
immediately
before
the
statement
of
this
Lemma.
Thus,
since
G
η
P
→
F
η
P
is
η
→
G
η
cannot
split
either.
This
contradiction
completes
birational,
it
follows
that
G
P
P
the
proof
of
the
Lemma.
Now
let
us
assume
that
we
are
given
a
surjective
continuous
homomorphism
over
Γ
K
θ
:
Π
U
K
→
Π
Y
K
def
where
Y
K
is
a
proper
hyperbolic
curve
over
K.
Let
Y
η
P
=
Y
K
×
K
η
P
.
Thus,
Y
η
P
is
a
proper
hyperbolic
curve
over
η
P
.
Suppose,
moreover,
that
we
know
that
morphisms
like
θ
necessarily
arise
geometrically
(i.e.,
from
a
dominant
morphism
U
K
→
Y
K
)
for
U
K
of
dimension
<
n.
(We
shall
refer
to
this
assumption
as
the
“Induction
Hypothesis.”
83
Note
that
by
Corollary
15.5,
we
already
know
that
this
induction
hypothesis
is
true
for
dim
K
(U
K
)
=
1.)
Then
the
morphism
Π
G
η
→
Π
Y
K
P
obtained
by
composing
(“π
1
of”)
the
natural
morphism
G
η
P
→
U
K
with
θ
lies
over
Γ
η
P
→
Γ
K
,
and
by
Lemma
16.2,
is
such
that
it
induces
a
surjection
between
the
(maximal
pro-
p
quotients
of
the)
respective
geometric
fundamental
groups.
Moreover,
this
morphism
naturally
defines
a
morphism
θ
G
:
Π
G
η
→
Π
Y
η
P
P
(since
Π
Y
η
is
the
fibered
product
of
Π
Y
K
and
Γ
η
P
over
Γ
K
).
Thus,
to
summarize,
θ
G
is
P
a
surjective
continuous
homomorphism
over
Γ
η
P
.
Let
U
η
G
P
be
the
generic
point
of
G
η
P
.
Thus,
by
composing
(“π
1
of”)
U
η
G
P
→
G
η
P
with
θ
G
,
we
obtain
a
continuous
surjective
homomorphism
θ
U
G
:
Π
U
η
G
→
Π
Y
η
P
P
over
η
P
.
Moreover,
the
dimension
of
U
η
G
P
over
η
P
is
<
n.
Thus,
by
the
induction
hypothesis,
it
follows
that
θ
U
G
arises
geometrically,
from
some
unique
dominant
morphism
U
η
G
P
→
Y
η
P
.
Projecting
to
Y
K
then
gives
a
dominant
morphism
U
η
G
P
→
Y
K
.
We
would
like
to
observe
that
this
morphism
U
η
G
P
→
Y
K
extends
to
U
K
.
Indeed,
to
see
this,
observe
that
U
η
G
P
and
U
K
are
both
projective
limits
of
open
subsets
of
the
projective
K
-variety
X
K
.
Moreover,
U
K
can
be
written
as
the
projective
limit
of
K
-
smooth
open
subsets
of
X
K
.
Thus,
the
fact
that
we
get
a
morphism
U
K
→
Y
K
follows
from
the
following
Lemma
16.3.
Let
L
be
a
field;
Y
L
be
a
proper,
smooth,
geometrically
connected
curve
of
nonzero
genus
over
L;
and
Z
L
be
a
smooth
L-variety.
Suppose
that
we
have
a
rational
map
φ
from
Z
L
to
Y
L
.
Then
φ
is
defined
over
all
of
Z
L
.
Proof.
This
Lemma
is
a
well-known
consequence
of
the
classical
theory
of
the
Albanese
variety
(see,
e.g.,
[Lang]):
Namely,
let
U
L
⊆
Z
L
be
a
(nonempty)
open
over
which
φ
is
defined.
Assume
(without
loss
of
generality)
that
Y
L
(L)
=
∅.
Let
A
L
be
the
Jacobian
of
Y
L
,
and
let
B
L
be
its
dual.
Then
a
point
of
Y
L
(L)
defines
an
embedding
Y
L
→
A
L
which
we
can
compose
with
φ
to
obtain
a
rational
map
φ
A
from
Z
L
to
A
L
.
It
suffices
to
show
84
that
φ
A
extends
to
a
morphism
on
Z
L
.
But
note
that
the
portion
of
φ
A
which
already
is
a
morphism
(from
U
L
to
A
L
)
defines
(by
pulling
back
the
Poincaré
bundle
on
A
L
×
L
B
L
)
a
line
bundle
on
U
L
×
L
B
L
.
Since
Z
L
×
L
B
L
is
a
regular
scheme,
it
follows
that
this
line
bundle
extends
to
Z
L
×
L
B
L
.
Then
taking
the
classifying
morphism
associated
to
this
extended
line
bundle
gives
a
morphism
Z
L
→
A
L
,
as
desired.
Let
us
review
what
we
have
done
so
far.
We
started
with
a
continuous
surjective
homomorphism
θ
:
Π
U
K
→
Π
Y
K
over
Γ
K
.
Moreover,
we
have
shown
that
if
we
base-change
θ
from
K
to
K
to
obtain
θ
K
:
Π
U
K
→
Π
Y
K
then
θ
K
arises
geometrically
from
some
dominant
U
K
→
Y
K
.
Note,
moreover,
that
this
morphism
U
K
→
Y
K
is
the
unique
morphism
that
gives
rise
to
(the
geometric
portion
of)
θ.
(This
follows,
for
instance,
from
the
inductive
hypothesis
on
n.
Moreover,
this
uniqueness
also
holds,
of
course,
over
any
finite
extension
of
K
.)
Thus,
if
we
specialize
the
indeterminate
t
∈
K
to
some
element
of
a
finite
extension
L
of
K
that
is
sufficiently
generic
so
that
U
K
→
Y
K
specializes
to
U
L
→
Y
L
,
and
then
base-change
this
U
L
→
Y
L
back
up
to
a
morphism
U
L
→
Y
L
(where
L
is
the
composite
of
K
and
L),
then
this
morphism
U
L
→
Y
L
coincides
with
the
morphism
obtained
by
base-changing
the
original
U
K
→
Y
K
via
K
⊆
L
.
But
this
means
that
the
original
U
K
→
Y
K
is,
in
fact,
defined
over
L.
Finally,
since
K
and
L
are
linearly
disjoint
over
K,
it
follows
that
the
original
U
K
→
Y
K
is
defined
over
K.
Thus,
we
get
a
morphism
U
K
→
Y
K
which
clearly
induces
the
original
θ
(since,
for
instance,
Π
U
K
→
Π
U
K
is
surjective).
Let
us
step
back
now
and
take
stock
of
what
we
have
done
so
far
in
this
Section.
We
started
with
U
K
,
the
spectrum
of
a
function
field
over
K,
and
a
proper
hyperbolic
curve
Y
K
over
K.
Then
given
any
surjective
continuous
homomorphism
θ
:
Π
U
K
→
Π
Y
K
over
Γ
K
,
we
showed
that
θ
necessarily
arises
geometrically.
Now
let
us
pause
for
a
defini-
tion:
Definition
16.4.
Let
Q
K
be
a
K-scheme.
We
shall
call
Q
K
a
smooth
pro-variety
if
it
is
the
projective
limit
of
a
projective
system
of
smooth
(geometrically
connected)
varieties
over
K
such
that
the
transition
morphisms
are
all
birational.
Now
Lemma
16.3,
plus
the
techniques
of
the
proofs
of
Theorem
14.1
and
Corollary
14.2
show
that
we
have,
in
fact,
proven
the
following
(our
first
main
theorem
–
Theorem
A
in
the
Introduction):
85
Theorem
16.5.
Let
K
be
sub-p-adic
(cf.
Definition
15.4
(i)).
Let
X
K
(respectively,
Y
K
)
be
a
smooth
pro-variety
(respectively,
hyperbolic
pro-curve)
over
K.
Let
Hom
dom
K
(X
K
,
Y
K
)
open
the
set
of
dominant
K-morphisms
from
X
K
to
Y
K
.
Let
Hom
Γ
K
(Π
X
K
,
Π
Y
K
)
be
the
set
of
open,
continuous
group
homomorphisms
Π
X
K
→
Π
Y
K
over
Γ
K
,
considered
up
to
composi-
tion
with
an
inner
automorphism
arising
from
Δ
Y
.
Then
the
natural
map
open
Hom
dom
K
(X
K
,
Y
K
)
→
Hom
Γ
K
(Π
X
K
,
Π
Y
K
)
is
bijective.
Remark.
Finally,
we
make
the
following
important
observation:
Note
that
given
any
pro-p
result
such
as
Theorem
16.5,
one
can
always
immediately
derive
a
corresponding
profinite
result
from
it.
(Here
by
“corresponding
profinite
result,”
we
mean
the
same
result,
except
that
“Π”
(respectively,
“Δ”)
is
replaced
by
“Π
prf
”
(respectively,
“Δ
prf
”).)
Indeed,
suppose
that
K,
X
K
,
and
Y
K
are
as
in
the
statement
of
Theorem
16.5,
and
let
prf
θ
:
Π
prf
X
K
=
π
1
(X
K
)
→
Π
Y
K
=
π
1
(Y
K
)
be
an
open
homomorphism
over
Γ
K
.
Then
note
that
θ
immediately
induces
an
open
homomorphism
θ
p
:
Π
X
K
→
Π
Y
K
.
Applying
Theorem
16.5
shows
that
θ
p
arises
from
some
φ
:
X
K
→
Y
K
.
Thus,
it
remains
only
to
show
that
the
morphism
induced
by
φ
on
fundamental
groups
coincides
with
θ.
But
this
follows
from
the
argument
preceding
Theorem
14.1:
namely,
we
consider
an
arbitrary
finite
étale
covering
Y
K
→
Y
K
.
Pulling
this
covering
back
via
θ
gives
a
finite
étale
covering
X
K
→
X
K
.
Moreover,
θ
induces
a
morphism
between
the
full
profinite
fundamental
groups
of
X
K
and
Y
K
.
Next,
observe
that
this
morphism
gives
us
a
φ
:
X
K
→
Y
K
(by
the
same
argument
as
that
used
to
construct
φ
from
θ)
which
(by
naturality
of
the
constructions
involved)
lies
over
φ.
Thus,
if
we
continue
this
procedure
for
arbitrary
finite
étale
coverings,
the
well-known
correspondence
between
fundamental
groups
and
categories
of
étale
coverings
shows
that
the
morphism
induced
by
φ
on
fundamental
groups
coincides
with
θ
up
to
composition
with
an
inner
automorphism
induced
by
an
element
of
Δ
prf
Y
.
This
completes
the
proof
of
the
“profinite
analogue
of
Theorem
16.5.”
Section
17:
Maps
Between
Higher-Dimensional
Function
Fields
Let
K
be
sub-p-adic
(cf.
Definition
15.4
(i)).
Let
L
and
M
be
function
fields
(of
arbitrary
dimension)
over
K.
(Thus,
in
particular,
we
assume
here
that
K
is
algebraically
closed
in
L
and
M).
We
denote
by
Γ
L
and
Γ
M
the
absolute
Galois
groups
of
L
and
86
M,
respectively.
In
this
Section,
we
would
like
to
show
how
to
derive
a
result
(Theorem
17.1,
which
is
stated
as
Theorem
B
in
the
Introduction)
like
Theorem
16.5
for
morphisms
between
L
and
M
(over
K).
Theorem
17.1.
Let
K
be
sub-p-adic
(cf.
Definition
15.4
(i)).
Let
L
and
M
be
func-
tion
fields
of
arbitrary
dimension
over
K.
Let
Hom
K
(Spec(L),
Spec(M))
be
the
set
of
K-morphisms
from
M
to
L.
Let
Hom
open
Γ
K
(Γ
L
,
Γ
M
)
be
the
set
of
open,
continuous
group
homomorphisms
Γ
L
→
Γ
M
over
Γ
K
,
considered
up
to
composition
with
an
inner
automor-
phism
arising
from
Ker(Γ
M
→
Γ
K
).
Then
the
natural
map
Hom
K
(Spec(L),
Spec(M))
→
Hom
open
Γ
K
(Γ
L
,
Γ
M
)
is
bijective.
Proof.
First,
recall
(cf.
the
Remark
following
Theorem
16.5)
that
any
pro-p
result
such
as
Theorem
16.5
always
implies
a
corresponding
profinite
result.
Now
we
use
induction
on
the
transcendence
degree
–
which
we
shall
henceforth
denote
by
dim
K
(M)
–
of
M
over
K.
When
dim
K
(M)
=
0,
the
result
is
vacuous.
When
dim
K
(M)
=
1,
the
result
follows
from
the
profinite
version
of
Theorem
16.5;
thus,
we
may
assume
that
dim
K
(M)
>
1.
Now
suppose
that
we
know
Theorem
17.1
to
be
true
for
maps
to
functions
fields
of
transcendence
degree
<
dim
K
(M).
Suppose
that
we
are
given
an
open
continuous
homomorphism
θ
:
Γ
L
→
Γ
M
over
Γ
K
.
Observe
that
just
as
previously
(e.g.,
in
the
proof
of
Theorem
14.1),
we
can
assume
without
loss
of
generality
that
θ
is
surjective.
Next,
observe
that
there
exists
a
function
field
P
⊆
M
such
that
0
<
dim
K
(P
)
<
dim
K
(M).
Moreover,
we
may
assume
that
P
is
algebraically
closed
in
M.
Thus,
we
get
a
surjection
Γ
M
→
Γ
P
.
Composing
θ
with
this
surjection,
we
obtain
a
surjection
θ
P
:
Γ
L
→
Γ
P
,
which,
by
the
induction
hypothesis
on
dim
K
(M),
we
know
arises
geo-
metrically
from
some
P
→
L
(over
K).
Moreover,
since
θ
P
is
surjective,
it
follows
that
P
is
algebraically
closed
in
L.
Thus,
we
may
regard
L
and
M
as
function
fields
over
P
.
Moreover,
dim
P
(M)
<
dim
K
(M).
Thus,
since
θ
:
Γ
L
→
Γ
M
is
(by
the
definition
of
θ
P
)
a
morphism
over
Γ
P
,
it
follows
from
the
induction
hypothesis
on
dim
K
(M)
that
θ
arises
from
some
M
→
L
over
P
(hence
also
over
K).
This
completes
the
proof
of
the
Theorem.
Remark.
Note
that
in
this
case,
we
needed
to
work
with
profinite
(not
pro-p)
fundamental
groups
because
the
operation
of
taking
the
maximal
pro-p
quotient
is
not
well-behaved
with
respect
to
fibrations:
i.e.,
if
we
had
replaced
Ker(Γ
M
→
Γ
K
)
with
its
maximal
pro-p
quotient,
we
would
run
into
trouble
because
it
is
not
clear
that
the
maximal
pro-p
quotient
of
Ker(Γ
M
→
Γ
P
)
injects
into
the
maximal
pro-p
quotient
of
Ker(Γ
M
→
Γ
K
).
87
Also,
we
needed
to
work
with
function
fields
(as
opposed
to
varieties)
because,
if,
for
instance,
Spec(L)
and
Spec(P
)
had
been
varieties,
the
fact
that
the
geometric
(over
K)
fundamental
group
of
Spec(L)
surjects
onto
that
of
Spec(P
)
does
not
necessarily
imply
that
the
morphism
Spec(L)
→
Spec(P
)
has
geometrically
connected
fibers.
Moreover,
if
the
generic
geometric
fiber
of
Spec(L)
→
Spec(P
)
has
several
distinct
connected
components,
it
is
not
necessarily
the
case
that
the
fundamental
group
of
any
of
these
connected
components
surjects
onto
the
geometric
(over
P
)
fundamental
group
of
Spec(M).
Indeed,
by
replacing
Spec(M)
by
a
smooth
projective
variety
Y
K
(of
dimension
≥
3),
and
Spec(L)
by
the
result
–
call
it
X
K
–
of
cutting
this
variety
with
a
generic
hyperplane
section,
then
the
inclusion
X
K
→
Y
K
induces
an
isomorphism
on
geometric
fundamental
groups
despite
the
fact
that
X
K
is
not
isomorphic
to
Y
K
.
Moreover,
note
that
such
a
counterexample
to
a
“variety
version”
of
Theorem
17.1
exists
even
if
Y
K
is
“hyperbolic”
(say,
a
product
of
proper
hyperbolic
curves),
in
which
case,
one
would
expect
X
K
(at
least
if
the
ample
line
bundle
used
to
cut
Y
K
to
form
X
K
has
sufficiently
high
degree)
to
be
“quite
hyperbolic,”
as
well.
Section
18:
Truncated
Fundamental
Groups
def
If
Δ
is
a
topological
group,
let
us
introduce
the
following
notation:
Δ{0}
=
Δ;
for
def
i
≥
1,
Δ{i}
=
[Δ{i−1},
Δ{i−1}].
Also,
let
us
write
“Π
i
”
(respectively,
“Δ
i
”)
for
Π/Δ{i}
(respectively,
Δ/Δ{i}).
The
purpose
of
this
Section
is
to
observe
that
much
of
the
theory
of
this
paper
continues
to
hold
to
a
large
extent
even
when
we
consider
morphisms
not
between
the
full
Π’s,
but
between
certain
truncated
versions
of
the
Π’s.
Our
first
main
result
is
the
following:
Theorem
18.1.
Let
K
be
sub-p-adic
(cf.
Definition
15.4
(i)).
Let
X
K
be
a
smooth
variety
over
K.
Let
Y
K
be
a
hyperbolic
curve
over
K.
Let
n
≥
5.
Then
every
continuous
open
homomorphism
θ
:
Π
nX
K
→
Π
nY
K
over
Γ
K
induces
a
dominant
morphism
μ
:
X
K
→
Y
K
whose
induced
morphism
on
fun-
damental
groups
coincides
(up
to
composition
with
an
inner
automorphism
arising
from
n−3
Δ
Y
K
)
with
the
morphism
Δ
n−3
X
K
→
Δ
Y
K
defined
by
considering
θ
“modulo
Δ{n
−
3}.”
Proof.
First
let
us
consider
the
case
where:
(i)
K
is
a
finite
extension
of
Q
p
;
(ii)
X
K
is
a
curve;
(iii)
Y
K
is
a
non-hyperelliptic
proper
hyperbolic
curve;
(iv)
n
=
3.
Then
note
that
the
theory
of
Sections
1
through
5
manifestly
only
involves
Δ
2
X
K
.
It
thus
follows
immediately
(via
the
help
of
a
technical
lemma
–
Lemma
18.4
below
–
necessary
in
order
to
assure
that
Lemma
7.3
goes
through
in
the
truncated
context)
that
to
prove
Proposition
7.4,
we
really
only
used
Δ
2
Y
η
.
Thus,
in
summary,
as
long
as
the
covering
Y
η
S
→
Y
η
S
(in
S
88
the
notation
of
the
discussion
preceding
Proposition
7.4)
arises
from
an
open
subgroup
of
Π
1
Y
K
,
we
can
apply
Proposition
7.4
to
Y
η
S
to
conclude
that
Y
η
S
admits
a
line
bundle
of
degree
prime
to
p.
Now
we
would
like
to
conclude
the
“preservation
of
relations.”
Note
that
Section
11
manifestly
only
involves
the
abelianization
of
the
geometric
fundamental
group.
More-
over,
Section
12
is
just
formal
manipulation.
In
Section
13,
the
only
thing
from
Sections
1
through
7
that
is
used
is
Proposition
7.4
(whose
applicability
under
the
present
circum-
stances
was
discussed
in
the
preceding
paragraph).
Thus,
by
using
Propositions
7.4
and
8.1,
Lemma
8.2,
and
the
theory
of
Section
9,
we
may
conclude
“preservation
of
relations”
for
the
morphism
H
0
(Y
K
,
ω
Y
K
/K
)
→
H
0
(X
K
,
ω
X
K
/K
)
induced
by
Δ
1
X
K
→
Δ
1
Y
K
.
(Note,
however,
that
this
time,
in
the
application
of
Section
9,
we
take
for
our
tower
of
coverings
“X
L
n
”
not
the
coverings
corresponding
to
the
p-derivates
of
the
whole
geometric
fundamen-
tal
group
Δ,
but
rather
the
p-derivates
of
Δ
1
–
which
is
enough
to
carry
out
the
argument
of
Section
9.)
Hence
we
get
a
morphism
μ
:
X
K
→
Y
K
.
Now
let
us
lift
the
hypothesis
that
n
=
3.
Then
we
obtain
the
following:
If
Y
K
→
Y
K
is
a
covering
arising
from
an
open
subgroup
of
Π
n−3
Y
K
,
and
X
K
→
X
K
is
a
connected
component
of
the
pull-back
of
this
covering
to
X
K
via
θ,
then
we
get
a
natural
morphism
μ
:
X
K
→
Y
K
.
Moreover,
the
naturality
of
the
construction
of
this
morphism
means
that
it
always
lies
over
the
morphism
X
K
→
Y
K
.
Thus,
it
follows
by
the
usual
argument
(i.e.,
the
one
preceding
Theorem
14.1)
that
the
morphism
induced
by
μ
on
fundamental
groups
is
equal
to
θ
modulo
Δ{n
−
3}
(up
to
composition
with
an
inner
automorphism
arising
from
Δ
Y
K
).
So
far
we
have
not
used
that
n
≥
5.
(In
fact,
we
have
only
used
that
n
≥
3.)
The
purpose
of
assuming
that
n
≥
5
is
to
lift
hypothesis
(iii)
(in
the
first
paragraph
of
this
proof).
Namely,
given
any
hyperbolic
curve
Y
K
,
it
is
elementary
to
show
that
there
always
exists
a
covering
Y
K
→
Y
K
defined
by
an
open
subgroup
of
Π
2
Y
K
such
that
the
compactification
Y
K
of
Y
K
is
hyperbolic
and
non-hyperelliptic,
and
such
that
Y
K
→
Y
K
has
arbitrarily
large
(specified)
ramification
over
all
the
points
of
Y
K
−
Y
K
.
Thus,
we
get
a
→
Y
K
which
descends
to
a
map
X
K
→
Y
K
.
Moreover,
the
fact
that
X
K
→
X
K
map
X
K
is
étale,
while
Y
K
→
Y
K
is
ramified
over
the
points
of
Y
K
−
Y
K
(with
ramification
indices
arbitrarily
large)
implies
that
the
map
X
K
→
Y
K
factors
through
Y
K
.
Thus,
we
get
a
map
X
K
→
Y
K
,
as
desired.
Then
arguing
as
in
the
preceding
paragraph
completes
the
proof,
albeit
still
under
the
assumptions
(i)
and
(ii)
(of
the
first
paragraph
of
the
proof).
The
extension
to
the
case
of
fields
K
that
are
subfields
of
finitely
generated
extensions
of
Q
p
then
follows
via
the
same
“specialization
argument”
as
that
employed
previously
in
the
nontruncated
case
(following
Lemma
15.2).
Thus,
we
can
also
lift
assumption
(i).
Finally,
the
“cutting
with
a
hyperplane
argument”
of
Section
16
extends
immediately
to
the
truncated
case.
This
allows
us
to
lift
assumption
(ii),
thus
completing
the
proof
of
the
Theorem.
Theorem
18.2.
Let
K
be
a
subfield
of
a
finitely
generated
field
extension
of
Q
p
.
Let
X
K
be
a
smooth
pro-variety
over
K.
Let
Y
K
be
a
hyperbolic
pro-curve
over
K.
Let
n
0
be
89
the
minimum
transcendence
degree
over
Q
p
of
all
finitely
generated
field
extensions
of
Q
p
that
contain
K.
Let
n
0
be
the
transcendence
degree
over
K
of
the
function
field
of
X
K
.
def
Let
n
0
=
n
0
+
2(n
0
−
1)
+
1.
Let
n
≥
3n
0
+
5.
Then
every
continuous
open
homomorphism
θ
:
Π
nX
K
→
Π
nY
K
over
Γ
K
induces
a
dominant
morphism
μ
:
X
K
→
Y
K
whose
induced
morphism
on
fundamental
groups
coincides
(up
to
composition
with
an
inner
automorphism
arising
0
0
from
Δ
Y
K
)
with
the
morphism
Δ
n−3−3n
→
Δ
n−3−3n
defined
by
considering
θ
“mod-
X
K
Y
K
ulo
Δ{n
−
3
−
3n
0
}.”
Proof.
The
reason
for
the
inclusion
of
the
extra
“3n
0
”
(i.e.,
a
“price”
of
three
steps
for
every
additional
transcendence
degree
that
is
used,
plus
an
extra
“tax”
of
three
steps
for
allowing
the
prefix
“pro”)
is
the
following:
in
order
to
reduce
to
the
situation
discussed
in
the
proof
of
Theorem
18.1,
we
need
to
show
(in
the
present
“pro”
context)
that
inertia
groups
are
annihilated.
Moreover,
to
apply
the
first
“inertia
annihilation
argument”
(at
the
beginning
of
Section
14),
we
needed
to
know
that
after
one
passes
to
some
covering
Y
K
→
Y
K
arising
from
an
open
subgroup
of
Π
n−3
Y
K
(i.e.,
we
wish
to
apply
the
arguments
of
Theorem
18.1
for
n
−
3),
one
knows
“preservation
of
relations”
for
Y
K
.
Thus,
already
one
needs
some
extra
padding
–
to
the
tune
of
three
steps
(necessary,
as
we
saw
in
the
first
two
paragraphs
of
the
proof
of
Theorem
18.1,
to
derive
“preservations
of
relations”
for
Y
K
).
(This
accounts
for
the
1
in
the
definition
of
n
0
.)
Moreover,
each
time
one
adds
a
transcendence
degree,
one
needs
to
apply
the
“inertia
annihilation
argument”
of
the
first
half
of
Section
15.
Thus,
by
the
same
line
of
reasoning,
one
needs
extra
padding
consisting
of
three
steps.
(This
accounts
for
the
n
0
in
the
definition
of
n
0
.)
In
Section
16,
one
uses
not
only
the
transcendence
degrees
inherent
in
K,
but
also
2(n
0
−
1)
auxiliary
transcendence
degrees
–
here
the
“2
=
1
+
1”
comes
from
the
transcendence
degree
of
“K
”
over
“K,”
plus
the
transcendence
degree
of
the
pencil
“ξ.”
(This
accounts
for
the
2(n
0
−
1)
in
the
definition
of
n
0
.)
Finally,
we
remark
that
although
in
Sections
14,
15,
and
16,
we
assumed
that
the
morphism
of
fundamental
groups
“θ”
was
surjective,
it
is
easy
to
see
that
this
assumption
is
merely
cosmetic,
i.e.,
is
inessential
and
serves
only
to
simplify
the
discussion.
This
completes
the
proof
of
the
Theorem.
Remark.
Thus,
the
essential
difference
between
Theorems
18.1
and
18.2
is
that
in
Theo-
rem
18.2,
we
allow
“pro-objects,”
at
the
cost
of
having
to
apply
“annihilation
of
inertia”
arguments,
which
require
us
to
use
larger
quotients
of
Δ
X
K
,
Δ
Y
K
(i.e.,
each
application
of
annihilation
of
inertia
costs
three
units
of
“n”).
Remark.
We
do
not
mean
to
pretend
that
the
estimates
in
the
above
two
Theorems
(e.g.,
the
“5’s,”
“n
0
,”
etc.)
are
the
best
possible.
Especially
if
one
is
willing
to
add
hypotheses
90
to
Y
K
,
it
should
not
be
so
difficult
to
improve
these
estimates.
The
point
of
the
above
two
Theorems
is
simply
to
illustrate
the
principle
involved.
We
conclude
this
Section
with
two
technical
lemmas
that
were
used
in
the
proofs
of
the
above
two
Theorems.
Lemma
18.3.
Let
X
K
be
a
proper
hyperbolic
curve
over
a
field
K
of
characteristic
zero.
Let
Δ
X
be
(as
usual)
the
maximal
pro-p
quotient
of
its
geometric
fundamental
group.
Let
def
Ξ
X
=
Δ
X
/[Δ
X
,
[Δ
X
,
Δ
X
]].
Then
the
natural
morphism
H
2
(Ξ
X
,
Z
p
(1))
Γ
K
→
H
2
(Δ
X
,
Z
p
(1))
Γ
K
(induced
by
the
quotient
map
Δ
X
→
Ξ
X
)
is
surjective.
Here,
the
superscripted
“Γ
K
”
denotes
“the
submodule
of
Γ
K
-invariants.”
Proof.
Let
us
first
consider
the
morphism
H
2
(Ξ
X
,
Z
p
(1))
→
H
2
(Δ
X
,
Z
p
(1))
To
do
this,
we
shall
use
the
Hochschild-Serre
spectral
sequence
for
the
quotient
Ξ
X
→
H
X
(where,
as
usual,
we
write
H
X
for
the
abelianization
of
Δ
X
).
Write
Ψ
X
⊆
Ξ
X
for
the
kernel
of
Ξ
X
→
H
X
.
Thus,
Ψ
X
may
be
identified
with
a
certain
well-understood
(cf.
Lemma
3.1)
quotient
of
∧
2
H
X
.
Consideration
of
the
E
2
··
-term
of
this
spectral
sequence
shows
that
there
is
a
natural
Γ
K
-equivariant
injection
of
the
cokernel
of
the
natural
morphism
(induced
by
the
quotient
∧
2
H
X
→
Ψ
X
)
H
1
(Ψ
X
,
Z
p
(1))
=
Hom
Z
p
(Ψ
X
,
Z
p
(1))
→
H
2
(H
X
,
Z
p
(1))
=
Hom
Z
p
(∧
2
H
X
,
Z
p
(1))
into
H
2
(Ξ
X
,
Z
p
(1)).
(In
other
words,
the
above
morphism
is
the
differential
(from
E
2
0,1
to
E
2
2,0
)
of
the
“E
2
pq
=
H
p
(H
X
,
H
q
(Ψ
X
,
Z
p
(1)))”-term
of
the
spectral
sequence.)
Moreover,
it
follows
from
Lemma
3.1
that
the
cokernel
of
this
natural
morphism
is
precisely
the
quotient
of
H
2
(H
X
,
Z
p
(1))
=
(∧
2
H
1
(H
X
,
Z
p
))(1)
given
by
the
(surjective)
cup-product
map
(∧
2
H
1
(H
X
,
Z
p
))(1)
→
H
2
(Δ
X
,
Z
p
)(1).
Thus,
in
summary,
we
have
a
Γ
K
-equivariant
diagram:
H
2
(Ξ
X
,
Z
p
(1))
⊇
Image(H
2
(H
X
,
Z
p
(1)))
→
H
2
(Δ
X
,
Z
p
(1))
in
which
the
arrow
is
bijective.
This
implies
the
assertion
stated
in
the
Lemma.
91
Lemma
18.4.
Let
K,
X
K
,
Δ
X
,
and
Ξ
X
be
as
in
Lemma
18.3.
Since
the
kernel
of
Δ
X
→
Ξ
X
is
a
normal
subgroup
not
only
of
Δ
X
,
but
also
of
Π
X
K
,
write
Π
X
K
→
Π
Ξ
X
K
for
the
quotient
of
Π
X
K
by
this
subgroup.
Then
the
natural
morphisms
2
H
2
(Π
Ξ
X
K
,
Z
p
(1))
→
H
(Π
X
K
,
Z
p
(1))
and
2
H
2
(Π
X
K
×
Γ
K
Π
Ξ
X
K
,
Z
p
(1))
→
H
(Π
X
K
×
Γ
K
Π
X
K
=
Π
X
K
×
K
X
K
,
Z
p
(1))
are
surjective.
In
particular,
if
η
∈
H
2
(Π
X
K
×
K
X
K
,
Z
p
(1))
denotes
the
first
Chern
class
of
the
diagonal
in
X
K
×
K
X
K
,
then
η
lies
in
the
image
of
H
2
(Π
X
K
×
Γ
K
Π
Ξ
X
K
,
Z
p
(1)).
Proof.
The
assertions
of
this
Lemma
follow
by
considering
the
consequences
of
the
sur-
jectivity
assertion
of
Lemma
18.3
for
the
Hochschild-Serre
spectral
sequences
associated
Ξ
to
Π
Ξ
X
K
→
Γ
K
;
Π
X
K
→
Γ
K
;
Π
X
K
×
Γ
K
Π
X
K
→
Γ
K
;
and
Π
X
K
×
Γ
K
Π
X
K
→
Γ
K
.
Section
19:
Injectivity
Result
In
this
Section,
we
prove
the
following
“pro-p
injectivity
part
of
the
so-called
Section
Conjecture”:
Theorem
19.1.
Let
K
be
sub-p-adic
(cf.
Definition
15.4
(i)).
Let
X
K
be
a
hyperbolic
curve
over
K.
Let
X
K
(K)
be
the
set
of
K-valued
points
of
X
K
.
Let
Sect(Γ
K
,
Π
X
K
)
be
the
set
of
sections
Γ
K
→
Π
X
K
of
Π
X
K
→
Γ
K
,
considered
up
to
composition
with
an
inner
automorphism
arising
from
Δ
X
K
.
Then
the
natural
map
X
K
(K)
→
Sect(Γ
K
,
Π
X
K
)
is
injective.
Remark.
The
so-called
“Section
Conjecture”
states
that
(for
instance
when
X
K
is
proper)
the
natural
map
in
Theorem
19.1
is
bijective.
(When
X
K
is
affine,
the
statement
must
be
modified
slightly.)
At
the
time
of
writing
(July
1996),
the
Section
Conjecture
has
not
yet
been
proven.
Proof.
(of
Theorem
19.1)
Let
H
X
be
the
abelianization
of
Δ
X
.
First,
note
that
by
replacing
X
K
by
some
finite
Galois
étale
covering
of
X
K
of
p-power
order,
we
may
assume
92
that
the
compactification
X
K
of
X
K
is
itself
hyperbolic
and
non-hyperelliptic.
Let
r
be
the
number
of
points
in
(X
K
−
X
K
)(K).
Let
Z
X
be
the
complement
of
the
diagonal
in
X
K
×
K
X
K
.
By
taking
the
second
projection
Z
X
→
X
K
,
we
may
regard
Z
X
as
a
family
(parametrized
by
X
K
)
of
smooth
hyperbolic
curves
obtained
by
removing
r
+
1
distinct
points
from
some
compactification.
Let
η
X
be
the
generic
point
of
X
K
.
Let
η
X
be
the
spectrum
of
some
algebraic
closure
def
of
K(η
X
).
Let
Z
η
X
=
Z
X
×
X
η
X
.
Then
(since
Z
X
→
X
K
is
a
family
of
smooth
hyperbolic
curves
obtained
by
removing
precisely
r
+
1
distinct
points
from
some
compactification)
we
obtain
an
exterior
Galois
representation
ρ
Z
:
Π
prf
X
K
→
Out(Δ
Z
η
)
X
Now
I
claim
that
ρ
Z
factors
through
Π
X
K
.
Indeed,
to
see
this,
note
that
if
H
Z
ηX
is
the
abelianization
of
Δ
Z
η
,
then
since
Δ
Z
η
is
a
pro-p
group,
the
kernel
of
X
X
Ξ
:
Out(Δ
Z
η
)
→
Aut(H
Z
η
X
)
X
is
itself
a
pro-p
group.
Thus,
it
suffices
to
show
that
Ξ◦ρ
Z
maps
Δ
prf
X
into
a
pro-p
subgroup
of
Aut(H
Z
ηX
).
But
now
observe
that
one
has
a
Galois-equivariant
surjection
H
Z
ηX
→
H
X
whose
kernel
is
either
0
or
Z
p
(1)
(depending
on
whether
r
=
0
or
r
>
0).
Moreover,
the
actions
of
Δ
prf
X
on
H
X
and
Z
p
(1)
are
trivial.
Thus,
it
follows
immediately
that
the
prf
Δ
X
-action
on
H
Z
ηX
is
by
unipotent
matrices,
so
Ξ
◦
ρ
Z
maps
Δ
prf
X
into
a
pro-p
subgroup
of
Aut(M
Z
),
as
desired.
This
completes
the
proof
of
the
claim.
Thus,
we
have
a
representation
ρ
Z
:
Π
X
K
→
Out(Δ
Z
η
)
X
Now
suppose
that
we
have
two
points
α,
β
∈
X
K
(K)
that
induce
the
same
element
φ
∈
Sect(Γ
K
,
Π
X
K
).
Let
Z
α
(respectively,
Z
β
)
be
the
pull-back
of
Z
X
→
X
via
α
(respectively,
β).
Thus,
Z
α
and
Z
β
are
hyperbolic
curves
over
K.
Moreover,
since
the
action
of
Γ
K
on
Δ
Z
α
(respectively,
Δ
Z
β
)
is
determined
by
composing
φ
with
ρ
Z
,
it
follows
that
there
exists
some
isomorphism
ψ
:
Δ
Z
α
∼
=
Δ
Z
β
such
that
(i)
ψ
is
compatible
with
the
respective
outer
Γ
K
-actions;
(ii)
ψ
preserves
and
induces
the
identity
between
the
quotients
Δ
Z
α
→
H
X
,
Δ
Z
β
→
H
X
.
Thus,
by
Theorem
16.5
(in
fact,
really,
all
we
need
is
Theorem
A
of
[Mzk2]),
it
follows
that
ψ
arises
from
some
isomorphism
Z
α
∼
=
Z
β
which
is
compatible
with
and
induces
the
identity
on
the
inclusions
Z
α
→
X
K
,
Z
β
→
X
K
.
But
this
clearly
implies
that
α
=
β,
thus
completing
the
proof
of
the
Theorem.
Remark.
Note
that
in
fact,
in
the
proof
of
Theorem
19.1,
we
really
only
used
the
weaker
results
of
[Mzk2]
–
i.e.,
we
did
not
need
to
use
Theorem
A
of
the
present
paper.
On
the
93
other
hand,
at
the
time
that
[Mzk2]
was
written,
the
author
was
unaware
of
Theorem
19.1,
which
is
the
primary
reason
that
Theorem
19.1
did
not
appear
in
[Mzk2].
Note
also
that
although
Theorem
19.1
implies
a
corresponding
profinite
result,
the
profinite
result
can
be
proven
much
more
easily,
by
using
the
“Kummer
exact
sequence”
for
the
Jacobian
of
X
K
.
Finally,
it
should
be
remarked
that
the
argument
employed
in
the
proof
of
Theorem
19.1
(as
well
as
the
Kummer
sequence
argument
just
mentioned)
have
been
well-known
for
some
time
(see,
e.g.,
[Naka2]).
The
main
reason
that
Theorem
19.1
was
included
in
this
paper
was
that
the
author
just
wanted
to
make
explicit
that
a
pro-p
injectivity
result
(such
as
Theorem
19.1)
could
now
be
proven.
APPENDIX:
A
Grothendieck
Conjecture-Type
Result
for
Certain
Hyperbolic
Surfaces
Section
a0:
Introduction
In
this
Appendix,
we
show
how
Theorem
A
(cf.
the
Introduction)
can
be
used
to
prove
a
Grothendieck
Conjecture-type
result
for
certain
types
of
surfaces.
The
surfaces
considered
are
families
of
(smooth)
hyperbolic
curves
that
are
parametrized
by
hyperbolic
curves
(cf.
Definition
a2.1).
We
call
such
surfaces
hyperbolically
fibred.
Our
notation
is
similar
to
that
of
the
rest
of
the
paper,
except
that
since
here
we
consider
only
profinite
(i.e.,
not
pro-p)
fundamental
groups,
in
this
Appendix,
we
will
write
Π
X
K
for
the
profinite
fundamental
group
of
X
K
:
def
Notation:
If
K
is
a
field
and
X
K
is
a
K-scheme,
we
denote
by
Π
X
K
=
π
1
(X
K
)
the
fundamental
group
of
X
K
(for
some
choice
of
base-point),
and
by
Γ
K
the
absolute
Galois
group
of
K.
Then
we
have
a
natural
morphism
Π
X
K
→
Γ
K
whose
kernel
Δ
X
⊆
Π
X
K
is
the
geometric
fundamental
group
π
1
(X
K
⊗
K
K)
(where
K
is
an
algebraic
closure
of
K).
Our
main
theorem
is
the
following:
Theorem
D.
Let
K
be
sub-p-adic
(cf.
Definition
15.4
(i)).
Let
X
K
and
Y
K
be
hyperbolically
fibred
surfaces
over
K.
Let
Isom
K
(X
K
,
Y
K
)
be
the
set
of
K-isomorphisms
(in
the
category
of
K-schemes)
between
X
K
and
Y
K
.
Let
Isom
Γ
K
(Π
X
K
,
Π
Y
K
)
be
the
set
of
continuous
group
isomorphisms
Π
X
K
→
Π
Y
K
over
Γ
K
,
considered
up
to
composition
with
an
inner
automorphism
arising
from
Δ
Y
.
Then
the
natural
map
Isom
K
(X
K
,
Y
K
)
→
Isom
Γ
K
(Π
X
K
,
Π
Y
K
)
94
is
bijective.
This
Theorem
is
given
as
Theorem
a2.4
in
the
text.
We
remark
that:
(1)
Theorem
D
above
is
(modulo
a
certain
technical
result
–
Lemma
a1.1)
essentially
derived
from
Theorem
A.
Note
that
one
needs
the
full
power
of
the
“Hom
version”
of
the
Grothendieck
Conjecture
for
hyperbolic
curves
(i.e.,
Theorem
A)
in
order
to
prove
an
“Isom-type
result”
for
surfaces
(i.e.,
Theorem
D
above).
That
is
to
say,
the
“Isom
version”
of
Theorem
A
of
the
present
paper
–
i.e.,
Theorem
A
of
[Mzk2]
–
is
not
sufficient
to
prove
Theorem
D
above.
Moreover,
even
with
the
“Hom
version”
of
the
Grothendieck
Conjecture
for
hyperbolic
curves
(i.e.,
Theorem
A),
at
the
present
time,
I
am
unable
to
prove
a
“Hom
version”
of
Theorem
D.
(2)
One
might
ask
whether
Theorem
D
can
be
extended
to
hyperbolically
fibered
varieties
(i.e.,
varieties
obtained
as
successive
fibrations
of
hy-
perbolic
curves)
of
higher
dimension.
The
problem
is
that
just
as
we
needed
(see
Remark
(1))
the
“Hom
version”
in
dimension
one
to
prove
an
“Isom-type
result”
in
dimension
two,
we
would
need
a
“Hom-type
result”
in
dimension
two
–
which
is
currently
not
available
–
in
order
to
prove
an
“Isom-type
result”
in,
say,
dimension
three.
Thus,
at
the
present
time,
we
are
unable
to
advance
beyond
dimension
two.
(3)
Since
Theorem
A
is
valid
in
the
pro-p
case
as
well,
one
might
ask
why
one
cannot
prove
a
pro-p
version
of
Theorem
D
above.
The
problem
is
that
the
process
of
passing
to
the
maximal
pro-p
quotient
is
not
well-behaved
with
respect
to
fibrations.
That
is
to
say,
if
X
K
→
X
K
is
a
family
of
hyperbolic
curves
parametrized
by
a
hyperbolic
curve
(as
in
Definition
a2.1),
then
the
maximal
pro-p
quotient
of
the
fundamental
group
of
the
geometric
generic
fiber
of
X
K
→
X
K
does
not
map
injectively
(in
general)
to
the
maximal
pro-p
quotient
of
Δ
X
.
Thus,
any
attempt
to
prove
a
pro-p
version
of
Theorem
D
by
means
of
the
techniques
employed
here
would
result
in
a
rather
unnatural
theorem.
(4)
Since
Theorem
A
admits
various
truncated
versions
(cf.
the
Introduc-
tion)
as
well,
one
might
ask
why
one
cannot
prove
a
truncated
version
of
Theorem
D.
The
problem
here
is
the
same
as
the
problem
that
arises
when
one
tries
to
prove
a
pro-p
result:
i.e.,
truncating
is
not
well-behaved
with
respect
to
fibrations.
(5)
To
a
slight
extent,
the
content
of
Theorem
D
above
overlaps
with
recent
results
of
H.
Nakamura
and
N.
Takao
(see
Theorem
A
and
Corollary
B
of
[NT]).
95
Section
a1:
A
Key
Lemma
In
this
Section,
we
prove
a
simple
technical
lemma
which
will
be
the
key
technical
ingredient
that
allows
us
to
extend
Theorem
A
to
the
case
of
surfaces.
Let
K
be
an
algebraically
closed
field
of
characteristic
zero.
Let
X
and
Y
be
hyperbolic
curves
over
K
(cf.
Section
0
for
a
definition
of
this
term).
Lemma
a1.1.
Let
φ
:
X
→
Y
be
a
finite
K-morphism.
Let
ψ
:
Π
X
K
→
Π
Y
K
be
the
induced
morphism
on
fundamental
groups.
Suppose
that
Ker(ψ)
is
topologically
finitely
generated.
Then
φ
is
étale.
Proof.
By
replacing
Y
by
the
finite
étale
covering
of
Y
corresponding
to
Im(ψ)
⊆
Π
Y
K
,
we
may
assume
that
ψ
is
surjective.
Under
this
assumption,
φ
is
étale
if
and
only
if
φ
is
an
isomorphism.
Thus,
it
suffices
to
assume
that:
(1)
Ker(φ)
is
topologically
finitely
generated;
and
(2)
φ
is
not
an
isomorphism
(hence
has
degree
>
1)
and
derive
a
contradiction.
def
Let
Y
→
Y
be
a
finite
étale
covering
Y
,
where
Y
is
connected.
Let
X
=
X
×
Y
Y
.
Then
X
→
X
is
finite
étale,
and
it
follows
from
the
assumption
that
ψ
is
surjective
that
X
is
connected.
Since
Y
is
hyperbolic,
it
follows
that
for
a
suitable
choice
of
Y
→
Y
,
Y
will
have
genus
≥
2.
Thus,
by
replacing
our
original
X
→
Y
by
X
→
Y
,
we
may
assume
that
X
and
Y
have
genus
≥
2.
Since
X
→
Y
is
assumed
to
have
degree
>
1,
it
follows
that
g
X
(the
genus
of
X)
is
strictly
greater
than
g
Y
(the
genus
of
Y
).
Let
Y
→
Y
be
a
finite
étale
covering
of
degree
d
such
that
Y
is
connected,
and
Y
→
Y
extends
to
an
étale
covering
over
some
compactification
of
Y
.
Then
the
Riemann-Hurwitz
formula
implies
that,
as
Y
→
Y
varies,
g
Y
(which
we
think
of
as
a
function
of
d)
is
equal
to
d(g
Y
−
1)
+
1.
On
the
other
hand,
def
if
X
=
X
×
Y
Y
,
then
(again
by
the
Riemann-Hurwitz
formula)
g
X
=
d(g
X
−
1)
+
1.
Thus,
it
follows
that
as
d
→
∞,
the
difference
g
X
−
g
Y
→
∞.
Now
let
H
X
denote
the
first
homology
group
of
the
compactification
of
X
with
coefficients
in
Z
p
(for
some
prime
number
p
which
will
be
fixed
throughout
the
discussion).
Then
ψ
induces
a
surjection
H
ψ
:
H
X
→
H
Y
.
Thus,
Ker(H
ψ
)
is
a
free
Z
p
-module
of
rank
2(g
X
−
g
Y
).
On
the
other
hand,
any
topological
generators
of
Ker(ψ)
clearly
define
a
set
of
Z
p
-generators
of
Ker(H
ψ
).
But
this
implies
that
rank
Z
p
(Ker(H
ψ
))
is
bounded,
independently
of
Y
→
Y
,
which
contradicts
the
fact
that
rank
Z
p
(Ker(H
ψ
))
=
2(g
X
−
g
Y
)
→
∞
as
d
→
∞.
This
contradiction
completes
the
proof.
96
Section
a2:
The
Main
Theorem
Let
K
be
a
field
of
characteristic
zero.
Let
X
K
be
a
surface
over
K,
by
which
we
mean
that
X
K
is
a
smooth
(geometrically
connected)
variety
over
K
of
dimension
two.
Definition
a2.1.
We
shall
say
that
X
K
is
a
hyperbolically
fibred
surface
if
the
following
condition
holds:
There
exists
a
hyperbolic
curve
X
K
over
K,
together
with
a
smooth,
proper,
connected
morphism
X
K
→
X
K
of
relative
dimension
one
such
that
X
K
embeds
as
an
open
subvariety
of
X
K
satisfying
the
following
conditions:
(i)
X
K
−
X
K
is
a
divisor
;
(ii)
the
geometric
fibers
of
X
K
⊆
X
K
→
X
K
are
hyperbolic
in
X
K
which
is
étale
over
X
K
curves.
If
X
K
is
a
hyperbolically
fibred
surface,
then
we
shall
refer
to
X
K
→
X
K
(as
above)
as
a
parametrizing
morphism
for
X
K
.
Note
that
given
a
hyperbolically
fibered
surface
X
K
,
in
general,
there
can
exist
more
than
is
a
parametrizing
morphism
for
one
parametrizing
morphism
for
X
K
.
If
X
K
→
X
K
X
K
,
and
F
Ω
⊆
X
K
is
a
fiber
of
this
morphism
(over
some
point
of
X
K
(Ω)
valued
in
an
algebraically
closed
field
Ω),
then
we
have
an
exact
sequence
of
fundamental
groups:
1
→
Π
F
Ω
→
Π
X
K
→
Π
X
→
1
K
are
hyperbolic
curves,
it
follows
that
Π
F
Ω
and
Π
X
(hence
Note
that
since
F
Ω
and
X
K
K
also
Π
X
K
)
are
topologically
finitely
generated.
Now
let
K
be
sub-p-adic
(cf.
Definition
15.4
(i))
(for
some
prime
number
p),
and
let
X
K
and
Y
K
be
hyperbolically
fibred
surfaces
over
K.
Let
φ
:
Π
X
K
→
Π
Y
K
be
a
continuous
group
isomorphism
over
Γ
K
.
We
would
like
to
show
that
φ
arises
geomet-
rically.
Let
ζ
Y
:
Y
K
→
Y
K
be
a
parametrizing
morphism
for
Y
K
.
Then
by
composing
φ
with
π
1
(ζ
Y
),
we
obtain
a
continuous
surjection
Π
X
K
→
Π
Y
.
By
Theorem
A,
it
follows
K
that
this
surjection
arises
geometrically,
from
some
morphism
X
K
→
Y
K
.
Thus,
we
may
regard
X
K
and
Y
K
as
objects
over
Y
K
.
Let
Z
K
be
the
normalization
of
Y
K
in
X
K
.
Thus,
the
morphism
X
K
→
Y
K
factors
through
δ
:
Z
K
→
Y
K
.
Observe
that
Z
K
is
a
smooth,
geometrically
connected
(since
X
K
is
geometrically
connected
over
K)
curve
over
K,
and
that
Z
K
→
Y
K
is
finite.
(In
particular,
since
Y
K
is
hyperbolic,
it
follows
that
Z
K
is
also
hyperbolic.)
Moreover,
it
follows
from
the
definition
of
Z
K
that
the
morphism
Π
X
K
→
Π
Z
K
induced
by
X
K
→
Z
K
is
surjective.
This
implies
that
Ker(π(δ))
is
a
quotient
of
Ker(Π
X
K
→
Π
Y
)
∼
=
Ker(Π
Y
K
→
Π
Y
),
which
(by
the
definition
of
a
hyperbolically
fibred
K
K
97
surface)
is
topologically
finitely
generated.
Thus,
it
follows
that
Ker(π(δ))
is
topologically
finitely
generated.
But
then
Lemma
a1.1
implies
that
Z
K
→
Y
K
is
étale.
On
the
other
hand,
since
Π
X
K
→
Π
Y
is
surjective,
we
thus
see
that
Z
K
→
Y
K
must
be
an
isomorphism.
K
In
particular,
we
thus
conclude
that
the
generic
fiber
of
X
K
→
Y
K
is
smooth,
geometrically
connected,
and
of
dimension
one.
In
fact,
the
argument
of
the
preceding
paragraph
can
be
applied
more
generally
to
coverings
of
X
K
,
as
follows:
Let
Y
K
→
Y
K
be
a
finite
étale
covering
such
that
Y
K
is
geometrically
connected
over
K.
Let
Y
K
→
Y
K
×
Y
K
Y
K
be
a
finite
étale
covering
such
that
K
→
X
K
be
the
covering
corresponding
(via
Y
K
→
Y
K
is
geometrically
connected.
Let
X
K
→
X
K
factors
through
X
K
→
X
K
×
Y
Y
.
Then
I
claim
that
φ)
to
Y
K
→
Y
K
.
Thus,
X
K
K
K
→
Y
is
geometrically
connected.
Indeed,
this
follows
by
the
same
argument
as
that
X
K
employed
in
the
preceding
paragraph:
Namely,
we
simply
observe
that
Π
→
Π
is
K
K
X
Y
isomorphic
to
Π
→
Π
,
which
is
surjective
with
topologically
finitely
generated
kernel.
K
K
Y
Y
This
proves
the
claim.
Let
L
be
the
function
field
of
Y
K
;
let
L
be
its
algebraic
closure.
Note
that
the
natural
map
Spec(L)
→
Y
K
factors
through
Y
K
.
Moreover,
the
above
claim
K
×
Spec(L)
K
→
Y
via
Spec(L)
→
Y
,
the
resulting
X
implies
that
if
we
base-change
X
K
K
K
Y
is
connected.
Now
let
us
reinterpret
the
conclusions
of
the
preceding
paragraph
in
terms
of
funda-
def
def
mental
groups.
Let
η
Y
be
the
generic
point
of
Y
K
.
Let
Y
η
Y
=
Y
K
×
Y
K
η
Y
;
X
η
Y
=
X
K
×
Y
K
η
Y
.
Then
what
we
did
in
the
preceding
paragraph
implies
precisely
that
the
morphism
Ker(π
1
(X
η
Y
)
→
π
1
(η
Y
))
→
Ker(π
1
(Y
η
Y
)
→
π
1
(η
Y
))
=
Ker(Π
Y
K
→
Π
Y
)
K
(induced
by
φ)
is
surjective.
Thus,
we
see
that
by
composing
the
natural
surjection
of
π
1
(X
η
Y
)
onto
Π
X
K
with
φ,
we
obtain
a
continuous
surjective
group
homomorphism
φ
η
Y
:
Π
X
η
→
Π
Y
η
Y
Y
over
Γ
η
Y
(where
we
regard
X
η
Y
and
Y
η
Y
as
curves
over
η
Y
).
Now
we
would
like
to
apply
Theorem
A
again.
This
Theorem
tells
us
that
φ
η
Y
arises
geometrically
from
some
η
Y
-morphism
X
η
Y
→
Y
η
Y
.
Remark.
Note
that
in
the
argument
of
the
preceding
three
paragraphs,
it
was
absolutely
essential
to
invoke
Lemma
a1.1
before
we
could
apply
Theorem
A.
In
fact,
if
the
variety
“on
the
left”
(i.e.,
in
this
case,
X
η
Y
)
is
not
geometrically
connected
over
the
base
field
(i.e.,
in
this
case,
L),
then
it
is
not
difficult
to
see
that
Theorem
A
does
not
hold.
Indeed,
to
construct
such
a
counterexample,
suppose
that
Y
K
is
proper,
and
let
H
K
⊆
Y
K
be
a
98
hyperplane
section
(with
respect
to
some
projective
embedding
of
Y
K
)
such
that
H
K
is
smooth
and
geometrically
connected
over
K,
and
π
1
(H
K
)
→
π
1
(Y
K
)
is
surjective.
(Such
an
H
K
exists
by
the
Lefshetz
hyperplane
theorem.)
Write
η
H
for
the
generic
point
of
H
K
,
and
take
for
our
“X
η
Y
”
any
hyperbolic
curve
over
η
H
.
Then
note
that
(if
H
K
is
sufficiently
generic
so
that
every
fiber
of
H
K
→
Y
K
contains
at
least
one
point
at
which
H
K
→
Y
K
is
étale,
then)
the
induced
morphism
π
1
(η
H
)
→
π
1
(Y
η
Y
)
is
also
surjective,
so
we
get
a
surjective
morphism
π
1
(X
η
Y
)
→
π
1
(Y
η
Y
)
(by
composing
the
(necessarily
surjective)
structure
morphism
π
1
(X
η
Y
)
→
π
1
(η
H
)
with
π
1
(η
H
)
→
π
1
(Y
η
Y
)).
But
this
surjective
morphism
does
not
arise
from
a
dominant
morphism
X
η
Y
→
Y
η
Y
.
This
completes
the
portion
of
the
proof
which
is
an
application
of
(the
nontrivial,
sur-
jectivity
part
of)
Theorem
A.
The
remainder
of
the
proof
will
consists
of
using
elementary
algebraic
geometry
to
show
that
the
morphism
X
η
Y
→
Y
η
Y
constructed
above
extends
to
a
morphism
X
K
→
Y
K
.
Lemma
a2.2.
The
morphism
X
η
Y
→
Y
η
Y
above
extends
to
a
morphism
X
K
→
Y
K
.
Proof.
First,
observe
that
there
exists
a
finite
étale
covering
V
K
→
Y
K
with
the
following
properties:
(1)
V
K
is
a
hyperbolically
fibered
surface
that
admits
a
parametrizing
mor-
phism
V
K
→
V
K
that
fits
into
a
commutative
diagram:
V
K
⏐
⏐
−→
V
K
⏐
⏐
Y
K
−→
Y
K
(2)
The
fibers
of
V
K
→
V
K
have
genus
≥
2.
Let
U
K
→
X
K
be
the
finite
étale
covering
that
corresponds
(via
φ)
to
V
K
→
Y
K
.
Let
V
K
→
V
K
be
the
family
of
proper
hyperbolic
curves
that
compactifies
V
K
→
V
K
(as
in
Definition
a2.1).
Write
η
V
for
the
generic
point
of
V
K
.
Note
that
there
exists
a
natural
map
U
K
→
V
K
covering
X
K
→
Y
K
,
and
that
both
U
K
and
V
K
are
geometrically
connected
over
K
→
Y
is
geometrically
connected).
η
V
(cf.
the
argument
used
above
to
show
that
X
K
Thus,
we
may
form
(geometrically
connected)
η
V
-curves
U
η
V
,
V
η
V
.
Moreover,
by
the
definition
of
U
K
,
we
have
a
commutative
diagram
U
η
V
⏐
⏐
−→
V
η
V
⏐
⏐
X
η
Y
−→
Y
η
Y
99
−→
V
η
V
Thus,
in
particular,
we
obtain
a
morphism
U
η
V
→
V
η
V
(over
η
V
).
I
claim
that
this
morphism
extends
to
a
morphism
U
K
→
V
K
.
Indeed,
by
the
elementary
theory
of
surfaces
–
“elimination
of
indeterminacy”
(see,
e.g.,
Theorem
5.5
of
Chapter
V
of
[Harts])
–
it
follows
K
→
U
K
(obtained
by
successively
blowing
up
smooth
points)
such
that
there
exists
some
U
that
the
birational
transformation
from
U
K
to
V
K
defined
by
U
η
V
→
V
η
V
extends
to
a
K
necessarily
maps
K
→
V
K
.
On
the
other
hand,
any
exceptional
curve
in
U
morphism
U
into
a
fiber
of
V
K
→
V
K
(this
follows
since
U
K
→
V
K
is
already
a
morphism).
Thus,
any
K
maps
quasi-finitely
into
a
fiber
of
V
K
→
V
(which
will
always
be
a
exceptional
P
1
in
U
K
smooth,
proper,
hyperbolic
curve)
–
which
is
clearly
absurd.
This
contradiction
completes
the
proof
of
the
claim.
Thus,
we
have
a
morphism
U
K
→
V
K
.
Let
Y
K
→
Y
K
compactify
Y
K
→
Y
K
(as
in
Definition
a2.1).
Then
clearly
V
K
→
Y
K
extends
to
a
finite
(in
general,
ramified)
morphism
V
K
→
Y
K
.
Thus,
to
summarize,
we
have
a
rational
map
from
X
K
to
Y
K
which
is
covered
by
a
morphism
U
K
→
V
K
,
where
U
K
(respectively,
V
K
)
is
finite
over
X
K
(respectively,
Y
K
).
By
elementary
algebraic
geometry,
this
implies
that
we
get
a
morphism
X
K
→
Y
K
(covered
by
U
K
→
V
K
).
To
complete
the
proof
of
the
Lemma,
we
must
verify
that
this
morphism
X
K
→
Y
K
factors
through
Y
K
.
To
do
this,
we
simply
choose
V
K
→
Y
K
in
the
above
argument
such
that
V
K
is
ramified,
with
very
large
(say,
compared
to
the
degree
of
X
K
→
Y
K
)
ramification
index,
over
all
of
the
divisor
Y
K
−
Y
K
.
(Note
that
such
coverings
exist,
by
the
exact
sequence
of
fundamental
groups
following
Definition
a2.1,
plus
the
well-known
fact
from
topology
that
the
fundamental
group
of
a
compact
surface
with
finitely
many
punctures
has
coverings
with
arbitrarily
large
ramification
index
at
those
punctures.)
Then
since
X
K
→
Y
K
is
covered
by
U
K
→
V
K
,
and
U
K
→
X
K
is
finite
étale,
we
see
that
we
obtain
a
contradiction,
unless
X
K
→
Y
K
maps
into
Y
K
.
Thus,
we
conclude
that
we
have
a
morphism
X
K
→
Y
K
,
as
desired.
Let
us
summarize
what
we
have
done
so
far.
We
started
with
a
continuous
group
isomorphism
φ
:
Π
X
K
→
Π
Y
K
over
Γ
K
.
From
φ,
we
constructed
a
K-morphism
X
K
→
Y
K
,
which
we
denote
by
α.
Since
φ
is
invertible,
we
thus
see
that
we
have
also
constructed
a
K-morphism
β
:
Y
K
→
X
K
from
φ
−1
.
Since
everything
we
have
been
doing
is
functorial,
it
follows
that
π
1
(β
◦
α)
is
the
identity
on
Π
X
K
(up
to
composition
with
an
inner
automorphism
arising
from
an
element
of
Δ
X
).
Thus,
by
Lemma
a2.3
below,
we
conclude
that
β
◦
α
is
the
identity
on
X
K
.
Similarly,
α
◦
β
is
the
identity
on
Y
K
.
Lemma
a2.3.
Let
γ
:
X
K
→
X
K
be
a
K-morphism
such
that
π
1
(γ)
is
the
identity
on
Π
X
K
(up
to
composition
with
an
inner
automorphism
arising
from
an
element
of
Δ
X
).
Then
γ
is
the
identity.
100
Proof.
First
observe
that
π
1
(γ)
is
trivially
compatible
(up
to
composition
with
a
geometric
.
Thus,
(the
injectivity
inner
automorphism)
with
the
map
induced
by
π
1
’s
by
X
K
→
X
K
part
of)
Theorem
A
tells
us
that
γ
is
compatible
with
X
K
→
X
K
.
Let
η
X
be
the
generic
point
of
X
K
.
Then
γ
defines
a
morphism
γ
η
X
:
X
η
X
→
X
η
X
which
induces
the
identity
(up
to
composition
with
a
geometric
inner
automorphism)
on
π
1
’s.
Applying
(the
injectivity
part
of)
Theorem
A
again
then
tells
us
that
γ
η
X
is
the
identity.
Thus,
it
follows
that
γ
is
the
identity,
as
desired.
Thus,
α
and
β
are
isomorphisms.
Moreover,
it
follows
from
the
construction
of
α
that
π
1
(α)
coincides
with
φ
(up
to
composition
with
an
inner
automorphism
arising
from
an
element
of
Δ
Y
),
and
it
follows
from
Lemma
a2.3
that
α
is
the
unique
such
K-isomorphism
X
K
∼
=
Y
K
.
Thus,
we
see
that
we
see
that
we
have
proven
the
following
result:
Theorem
a2.4.
Let
K
be
sub-p-adic
(cf.
Definition
15.4
(i)).
Let
X
K
and
Y
K
be
hyperbolically
fibred
surfaces
over
K.
Let
Isom
K
(X
K
,
Y
K
)
be
the
set
of
K-isomorphisms
(in
the
category
of
K-schemes)
between
X
K
and
Y
K
.
Let
Isom
Γ
K
(Π
X
K
,
Π
Y
K
)
be
the
set
of
continuous
group
isomorphisms
Π
X
K
→
Π
Y
K
over
Γ
K
,
considered
up
to
composition
with
an
inner
automorphism
arising
from
Δ
Y
.
Then
the
natural
map
Isom
K
(X
K
,
Y
K
)
→
Isom
Γ
K
(Π
X
K
,
Π
Y
K
)
is
bijective.
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